I travel a lot and often find grand pianos in lobbies of hotels. Is it appropriate to just sit down and play them for a while (FWIW, I am good and play slow classical music. My playing will not ruin the mood)? Or are they primarily for show?
Every piano I've ever seen has had a 'not for public use' (or equivalent) sign on it. If you find one that doesn't it's still polite to ask, but they're probably content with people playing it if they don't say otherwise. Or they just installed it this morning and haven't seen the problem yet ...
If you can play the piano decently I really doubt anyone will mind, or if they do they'll politely ask you to stop. The general problem -- which is why they don't let anyone play -- is people just either repeatedly tapping out the one tune they know or banging random notes and throwing it out of tune.
Asking is always the best policy, BUT if you're actually good at playing and play something situation appropriate in my experience most places will turn a blind eye -- they're getting free 'mood music' essentially. The only problem is that it encourages other people to try their hand which may be less successful.
I couldn't play piano to save my life, but I know people who can and have seen them play hotel pianos without any incident.
I would feel comfortable sitting down at any piano that doesn't have a sign saying not to. Assuming they don't have music playing that you would be in competition with, and assuming that you're good and aren't going to play the same thing over and over, I would keep the following principle in mind:
It's easier to ask forgiveness than permission. - Rear Admiral Grace Hopper (9 December 1906 – 1 January 1992)
and know that it's quite unlikely that you'll be asking forgiveness.
If there is a sign or background music, and you're not comfortable taking the liberty, tell the person at the front desk that you have a few things you'd like to rehearse, and won't play indelicately, and you'll be very likely to get a smile and nod.
This is me in Vienna sitting down at an otherwise unused piano, and I've done the same at many hotels and building lobbies:
And at our most recent hotel stay, I did it again, and our friends are still commenting on it.
As said in the comments, ASK.
Pianos are expensive instruments, having every punter sit down and bang at them isn't good for them as you no doubt know.
I know the one in the lobby of the local hospital here is definitely NOT meant for visitors and patients to use, is reserved for organised performances (which happen several times a week). Of course it has a sign on it stating that, as the lobby is open 24/7 but the reception desk manned only during office hours.
Unless it's explicitly forbidden by a sign, I wouldn't see a good reason not to play. If you want to make sure, you can always ask. I've seen this kind of public pianos in hotels and in hospitals, and staff were always fine with people playing, at leats at daytime.
My main rationale for assuming that the piano may be used (which hasn't been mentioned in any of the answers so far) would be that as a hotel, they provide amenities such as a swimming pool and a tabletennis-table. Guests are allowed (or even supposed) to use those. It would be rather rude on the part of the hotel to favour sportive guests but deny musical guests the use of equivalent equipment.
And of course, by far not everyone who uses the sports facilities is proficient in them. Personally, I love playing tabletennis, I just cannot stand the kill-joys who insist the ball must not touch the floor or the walls. Hence, I'd see no reason why you'd even have to be confident to be playing well when you use such a public piano. As a matter of fact, for people who cannot play, a public hotel piano could be a rare chance to try, and for those who do play the piano in general, the hotel piano is their chance of not letting their regular training slide.
Lastly, all guests leave again sooner or later. If there are truly so many guests that want to use the piano despite not possessing too great skills at it that the hotel staff are bothered by it, the hotel should start rethinking whether their current placement of the piano is compatible with their actual target audience that visits their hotel.
In the Netherlands, NS, the national rail company, has been putting pianos in some of their busiest stations lately. They are always accompanied by a "Graag spelen mij" sign. I think it roughly translates to: please play me. On many occasions I have seen travellers sit down and play. And almost every time I notice a crowd gathering around the piano to listen.
I can't play the piano, but if I could do it reasonably well, I would not be able to pass-up such a chance. Unless stated otherwise, I think it is fair to just play. If you are unsure, asking is easy.