Is there any level limit for using gems?

Shortly and simply: no restrictions. :)

Quotes from wiki, who quoted blue posts:

They don’t have a level requirement so we do intend to see them used as a way to twink new characters, or allow people to buy into gemming up a bit earlier on if they have the gold.

There may be some interesting complications with this though.

You can pay an artisan to remove the gems from an item. The last design I had heard of was that it was based on gem value, so as you socket higher level gems it becomes more expensive, but you’ll almost certainly want to unsocket gems to level them up, or swap to new gear. This may cause some unique problems for low level (non-twink) characters attempting to buy high level gems and then being unable to remove them from the socket when they get a better piece of armor (because they can’t afford it), but that may turn out to be an acceptable roadblock.

Source: http://www.diablowiki.net/Gem

The simple answer is that there are no restrictions. But so far I believe the lowest level item was lvl 15 that contained a socket. So in theory this would be the only restriction due to the armor not the gem. As for removing gems, you can salvage the gear to get the gem back free of charge. You only lose the gear itself.