Physical Danger
There is nothing positive we could say about cancer. However, it does lack one attribute that we can safely say is really awesome: the Big C isn't contagious (source). To contract cancer from organs or tissue, you would literally have to put the organs or tissue inside of you, in which case you might not be qualified for this position.
There is no chance that you will ever "catch" cancer from studying it. There are dozens of reasons a person might get cancer, but touching it isn't one of those.
That doesn't mean that the laboratory is a danger-free zone. You'll be working with organic materials, so you'll want to wear proper safety equipment to protect you from other biological hazards.
Some labs will have chemicals, and you don't want to get any of that stuff in your eyes. Much of the lab equipment is large, expensive, contains pieces of metal and/or glass, and possibly heats up to 1,000 degrees, so make sure you keep your footsteps small and purposeful.
Thankfully, you'll learn all about proper lab research safety in college—either from a smart professor who teaches you, or a less-smart lab partner whose experiment puts you in the hospital for a week. Either way, lesson learned.