Is it legal to use rented dedicated servers to mine? I've got several dedicated servers which I'm renting for about 250 Euros from a big hoster (OVH). These server are used for things like ssh/scp, SVN and Git servers (for private repositories), HTTP proxy, etc.

Nothing shaddy here: just fully legal servers that see little use.

I realize these servers are really not powerful at all and would make terrible miners (at least for all the cryptocurrencies where GPU and ASIC mining is disproportionally powerful compared to CPU mining): that is not the point of my question.

My question is: can you legally use a dedicated server which you fully pay for to mine cryptocurrencies all year long? Basically I realized that the CPUs would be running at full speed and would consume more electricity than the way I use them now.
It's definitely not against the law. It might be breaking the terms of service (ToS) but there's really no reason why any dedicated server hosting provider wouldn't allow you to max out the CPU. I wouldn't do business with them if they limited that. However, you need to read your hosting provider's ToS. They might forbid certain applications like IRC servers, BitTorrent, Tor, etc. If they don't list Bitcoin then you are good to go.