Problem 2.20 (attributed to Lima) in Kirby's list of unsolved problems in low-dimensional topology asks:

Are there commuting homeomorphisms of the $2$-ball $B^2$ without a common fixed point?

What is the current status of this problem?

The answer is positive for higher-dimensional disks: Oliver (see here, page 175) has shown that there is a smooth action of $\mathbb Z_{30}\times\mathbb Z_{30}$ without fixed points on some disk of sufficiently high dimension.

The question also seems vaguely related to this one, about coincidence points of commuting self-maps of the disk. As stated there, such homeomorphisms are easily seen to have a coincidence point.