I think this is a situation where you need to use your brain a bit and use some of your best judgement.
Some might ask, "well if i go and download a single torrent and it takes 30 minutes and then I am done, will this be a problem?" In most cases, probably not as you hopefully will not be using up resources for a long period. However, downloading a torrent and then seeding it, especially for the not so legal torrents, is both abusive to the node and the network. When you purchase a VPS in the low end, you are getting a "Fair Share" of the resources (network and disk io) so anything that is going to thrash these things for long periods is not good. There is also the legality side of this... If you happen to get 1 DMCA warning and it seems to be an accident, I am sure they will work with you. If you decide to host the new Star Trek movie and allow a bunch of people to download it from you and get on someones radar and they start getting a bunch of DMCA complaints, are they going to put up with that? No. In most cases when using OpenVZ another thing to consider is the provider can directly see all your files, with little effort and if they receive a DMCA for something and the files are sitting there in your VPS, then obviously there is a problem and you have violated their TOS.
Now, the other question was, "What if I use a VPN?" Well, this is defiantly a better option than storing it directly on your server and abusing the io on the server, but you are still using the providers ip addresses and if they get more than 1 DMCA weather they can prove it or not they will most likely not want to keep you as a customer. If you are using a VPN, which is usually encrypted, the provider will not usually care about what you are doing as it is your business until they get a complaint. Is it possible to set realistic restrictions on your bit-torrent client and get away with downloading torrents through it, sure it is possible. Will any provider tell you that they allow this? No. For two reasons, #1 you could be downloading something that isn't legal and #2 Plausible deniability in a case where there is a DMCA or other complaint.
Tl;DR:
Use common sense, if you wouldn't download it directly to your home computer, don't do so on your VPS.
Can you get away with it if you are careful? Sure. Will a provider condone you doing this? Of course not. Is it wise to run a bit-torrent client on a vps server with shared resources? No, especially if you do not how to limit the resources the client is using. Is it smart to store illegal material on an OpenVZ VPS where the provider has direct access to your data in a legal situation? I think not. Are there ways to secure your data from your provider? With KVM it is possible. Would I suggest downloading torrents even if your data is secured on their network? Once again, not if you do not know what you are doing and do not know how to configure your resource usage appropriately. Can you get away with downloading torrents on VPN? It is possible. Is it a good idea? Use the above information to make an educated decision.