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It seems likes this is mostly an issue of scale, in well-traveled areas where lots of people go it's not going to make a lot of difference, while remote areas that could possibly see lots of sudden traffic, it would not be a good thing.

Also was one to be an "outdoor influencer" with thousands and thousands of followers rather than an average person, the effects could be that much more severe.

On the one hand, there’s the notion that posting content on the outdoors inspires others to get outside (see: #OptOutside, et al.). On the other, there’s the very real fear that posting photos of hidden hikes and hot springs invites an influx of visitors these places lack the resources to handle.

Geotagging can also get specific, and that’s where the real issues start. “We’re having a lot of problems with people geotagging hidden or sensitive places,” Boué said, adding that these places don’t always have the infrastructure to handle a lot of new visitors.

Source

There are any number of small specific things in the outdoors that are preserved because not many people know about them, geotagging hurts that preservation.

My general experience is that the problem is really that most tourists are focused in some very tiny but extraordinary popular areas. For example I have seen places in the Rockies where you need to book your camping place slots months in advance... But if you simply visit the next valley over (literally, distance was probably some 5-10km) you can walk for a week without meeting another soul. The same in Scandinavia... If you're walking Kungsleden for example you might encounter 100 people a day - go to any of the side trails and they will often be abandoned.

The popularity of already popular tourist destinations has certainly increased with the advent of social media and the readily available digital pictures. However, in my estimate this is rather one of these 'the rich get richer' (or here: the popular get more popular) situations...

I don't think that there is any immediate danger for the remote places you visit and then tag in pictures - unless you do some stunning shots of a really extraordinary place and they go viral.

If you are on a photo safari where you take pictures of endangered big-game animals (elephants, rhions, lions, etc.), do not geo-tag your images. Poachers are known to check the geo-information from such pictures in order to locate the territories of these animals and hunt them.

According to this article, signs like this can be found in wildlife reserves in South Africa:

Another situation where you should avoid geotagging is if you are taking a picture which shows you engaging in some activity in a place where this activity is prohibited. But you would of course never camp, fish, hunt, climb, drive, or build a fire at a location where it isn't explicitly allowed, would you?

There's a saying in Arizona "If you like our wildness, go away".

I'm of a different mind. By going into nature, and coming out of it wishing it to be protected, we do a service to it. How can we know how important nature is if we don't go into it? If you geotag a location, and more people see that it's a nice spot, there will be more effort to keep it the way it is. This will help to stop things like clear-cutting or mining from happening.

With that in mind, Charlie Brumbaugh's answer brings up a good point: There are already rules in some places. We should probably follow them.

As an answer, get in contact with the local wildlife people. What are their actual concerns? They might want more tourists.

I agree with most of the posters here. You're probably not going to have a hugely detrimental effect unless one of your photos goes viral and then it's a balance between encouraging people to get out and love nature and the effect that LOTS of people doing that at once will have. Generally remote areas are mostly safe from that as most people can't up go take a three-day hike at a moment's notice.

But as was mentioned with the Rhino, DO NOT geotag any endangered species - plants, animals, fungus, or lichen! If you see something really cool, maybe give it a quick google before posting and when in doubt, ask the management authority (though I can only speak to US authorities here). If it's federal land, ask the National Park Service or the Forest Service, for States, the Department of Natural Resources or State Fish and Wildlife agencies are good. Rangers at both levels tend to be really nice people that will be thrilled to talk to someone who just wants to help (as opposed to yell at them for something out of their control). For more local areas, try your county Extension agent (Extension programs are run through the State's land grant University), and they should at least be able to point you in the right direction.
Also, I imagine most of these people who you call will have a good beat on the land you photographed and have an opinion on whether geotagging could be detrimental.

Seriously though, I know it sounds lame, but you wouldn't believe how many people steal endangered plants. It really bums me out.

If you geotag a place that is easy to get to, you may have a detrimental influence. However, if you geotag a place that takes a long hike to get to, especially off-trail, multiday, and uphill, you are likely to have only a zero to very small impact.

I base this conclusion on our experiences in the California Sierra -- even Yosemite -- and the Colorado Rockies, where we could go days without seeing anybody, even in the distance, if we were off-trail and a thousand or more feet above the trailhead. Even more modest hikes attract few people beyond an obstacle that looks difficult (but is not).

A comment from a local quoted in the chapter The Moab Treehouse in Escape Routes by David Roberts is pertinent:

The legacy of secret-keeping is what gave us Glen Canyon Dam. "The
Place No One Knew."

Glen Canyon was a magnificent complex of branching canyons in Utah and Arizona. It was little known -- not remotely as famous as the Grand Canyon, hence the title of the book The Place No One Knew by Eliot Porter. The Grand Canyon was saved from being inundated by a dam because of widespread public outrage, but Glen Canyon, known only to a very few, was not. Boating on Lake Powell near the tops of the largely submerged canyons is, for many, a poor tradeoff. Edward Abbey wrote a novel The Monkey Wrench Gang, whose target was a thinly disguised Glen Canyon Dam.

I think it heavily depends on how one such place is administered.
There is a thing called 'footfall in nature' and unfortunately with a poor administration of people coming in it is likely to do more harm than good.

So, as an individual, I do not geo-tag anything that I find undisturbed. You might feel that I am a self-centred guy who want people to stay away from such hidden but beautiful places. If you feel that way, you are right about me.

Based on a decade-long experience in trekking in Western Ghats of India, looking at the current situation, I feel the less the people venture into nature, the better it is.

Case 1: Devkund trek banned for trekkers for the entire season

Devkund is a very scenic waterfall about 100 km away from two major metro cities of India. It lies the region which is an integral part of a major irrigation and hydro-electricity project in the state. A decade ago when I went there for the first time, we were only 6 people at the place. Recently, I saw the same place with about 1000 people, dancing around, playing loud music, littering everywhere, consuming alcohol like there is no tomorrow.
This change is worst for a fragile ecosystem. Although local residents are making decent money out of the whole thing in the trending season, authorities have no fallback plan to control the amount of footfall affecting the region.

The only fallback plan they could think of is: Police want to stop trekking in area

Case 2: Seasonal trekkers venturing out like frogs in rain.

Death toll rises during monsoon

In monsoon, on a particular trek usually a 1000 to 2000 trekkers are present on a given day. Most of them are not aware of 'Leave No Trace', most of them are hooligans. This situation is majorly due to Poor administration at places, uncontrolled formations of commercial trekking groups WITHOUT any regulation and compliance check. Every Tom-Dick-Harry forms a trekking organisation based on Facebook and Instagram, gathers a hundred people based on his/her experience of 10 treks last season. Geo-tagging, GPS tracks is their primary plan. On an average a group usually bring about 100 clients. They are hardly 4-5 people guiding(?) 100 people up the mountain. This is a recipe for disaster.

On the optimistic hand I see thing this way:
I never go out on a plan to discover one such place, which will elude people. I know my region well based on a decade-long (short) trekking experience, and yet I had to explore leaving a well-marked trail to get to some of the magical places. Those remain magical (not literally) because there aren't any hooligans visiting those places in hundreds of numbers at a time. I'd rather suggest not to disclose the location by any means.
Trekkers, who are experience enough, who have worked hard enough to know the region well, they will eventually get there, just like I did :-)

So, in a nutshell, as long as I am in India, I'll avoid geo-tagging purely because authorities and administrators fail miserably to control how many people visit the place, and basically how disciplined the visitors are.

Vox recently answered your question here - the youtube video is worth checking out

Horseshoe Bend used to be a little-known roadside view of the Colorado
River in Page, Arizona. But if you look at yearly satellite images of
the trail that leads to the river, you can see the spot undergo a
pretty dramatic transformation. In 1992, the trailhead is an empty
dirt patch — and in 2018, it consists of two adjacent parking lots
overflowing with cars and buses.

This spot went from a local secret to a viral tourist destination. The
main culprit for that uptick? Instagram.

The social media platform is reshaping the ways we buy things: our
clothes, our food, our products — and, via geotags, it’s also playing
a huge role in helping people decide where they want to travel.
Horseshoe Bend is now one of many hidden natural treasures across
America that have become too popular for their own good. And these
spaces that become popular after being geotagged often require
extensive redesign to protect visitors and the environment.

Digital popularity is resulting in physical changes in what these
places look like — and it’s raising serious concerns about how crowds
and construction can damage the natural landscape.