The main issue is accepting incoming connections. When you are behind a NAT (as most VPNs are for IPv4) you need some solution (such as port-forwarding) to make your torrent client connectable. This causes a number of issues when torrenting.

When someone starts a download they will try to connect to the seeders. If the seeders are not connectable this will fail.
As a fallback when the seeders notice the leachers they will try to connect to them. If the leacher also isn't connectable this will also fail.

If neither party is connectable the download can't happen, so you may fail to get content that you want.

This is extra relevant if you are on private trackers where seeding is tracked, has direct value and is competitive. If you are not connectable every new downloader will immediately connect to the connectable seeders and finish the download before your client even knows that they exist. (reannounces for seeders can be very infrequent, such as hourly, so it will take an average of 30min for you to notice a new seeder and try to connect to them). This makes it very difficult to acquire much upload unless there are very few other seeders.

NAT is evil, all hail IPv6.