I think every situation is different depending on the nature of the friendship between the chavrusas and how sensitive the person is. But I will tell you some things that I have seen done:
I had a friend breaking up with a very sensitive chavrusa. He (my friend) happens to be a very funny guy. One day, with as much obviously fake pomp as he could muster, he said to the chavrusa "It's not you - it's me." The humor of the bad cliche broke the tension, and the chavrusa he broke up with actually told me about it with a positive association because of the humor. That worked for him, probably won't work in every situation.
If you are in a situation where you could switch subjects and are comfortable doing so, you could say that you are switching topics and therefore won't be able to continue. If you are worried that he may be willing to switch along with you, you could line up a chavrusa before you tell him. I once had a friend who switched what shiur he was in in order to switch chavrusas. This depends on how flexible you are in your learning material.
When I broke up with chavrusas, I was very straight-forward, although I tried to find things to say that weren't objectively negative about him. I would point out differences in our learning styles or preferences, and perhaps point out that we as a unit were not getting so much done. This worked for me, but I always had very good friendships with my chavrusas.
I had a Rebbe who told me the following account, although you have to have the patience for it - every time his chavrusa would walk out, or fall asleep, or space out or whatever, he opened up a very small masechta (I think his choice was Megillah, but you could pick a smaller one). When he made a siyum, his chavrusa got the message that he needed to shape up.