Aside from their historical or ornamental value as furniture, old pianos are pretty useless.
The design and construction of string instruments like violins is pretty much a solved problem. Yes there is variation in the way the wood is shaped and carved, which gives you the different grades of quality, but the basic engineering problem of "how do you make a violin" is solved.
Because a piano is much more complex there is a lot of room for continual innovation in both materials and construction. This might be the reason that you don't see people performing on old pianos - the newer ones are just technologically superior. That and it just takes so much effort (and money) to maintain a quality piano.
That being said, there is a legion of both pianists and violinists that will tell you (incorrectly or not) that the sound of an instrument improves with age. Something about the fibers in the wood being affected by vibration.
[Edit: maintenance cost as a factor]
It has lost a lot of volume, some keys are "bited" and some strings get easily out of tune. But, believe me, it still sounds great. The lower range of notes beats most pianos I have heard (and played). It needs maintenance, but it's not useless. I think a piano can last a lot of time if you take care of it.
[1] http://www.pnas.org/content/early/2012/01/02/1114999109
More entertaining version: http://www.npr.org/blogs/deceptivecadence/2012/01/02/1444828...
I don't understand. Violins are instruments too. Did you just mean pianos, or others too?
A decently constructed instrument in the violin family will, as will (to a lesser extent) brass instruments.
I don't have a particular set of citations, it's just musician lore.