After a few visits, Mitch starts bringing a tape recorder with him to Morrie's house because he realizes that his time with Morrie is limited. As he explains to his friend:
I want to remember what we talk about, I told Morrie. I want to have your voice so I can listen to it… later. (10.2)
The tape recorder represents a certain mindfulness that comes to Mitch after spending some time with Morrie. Instead of just cycling through his days—wash, rinse, repeat—Mitch is tuning into them, and part of this is appreciating how fleeting his time with his friend is.
The tape recorder also shows Mitch's literal side. By recording his visits with Morrie, Mitch can go back and pick up on things that he didn't notice the first time. And, of course, by recording his voice, Mitch has a way to access Morrie sort of directly after he's gone. It is a gesture filled with love and admiration, a declaration of how much Morrie means to Mitch.
And lest you think this is just another way that Mitch avoids being vulnerable—he is trying to avoid losing Morrie completely once he dies, after all—consider this: He is putting memories that mean a whole heckofa lot to him on a piece of machinery that could totally break or somehow get lost. Looked at this way, the tapes are a risk, a way that Mitch dares to allow his feelings to emerge, just as much as they're a way of trying to shore up his access to his dear friend for the future.