31 dominos can be placed
Thanks to @Gamow's comment, this number's maximality can be proved
by self-contradiction of the assumption that it is not maximal.
Any more dominos would cover all 64 squares.
Assumption to be disproved: All squares can be covered with dominos.
A. As the top left corner must be covered, start with a horizontal domino
there. (Every other possible corner domino is equivalent to this
by rotation and reflection.) From here on:
Successive placements of a diagonal series of dominos
are forced to form a descending herringbone staircase in order to prevent
a 2x2 square from being formed in combination with each previous domino.
B. The domino that neighbors A along the left edge must be vertical.
C. The inside corner formed by A and B must be covered by
a horizontal domino.
D −  M. Likewise,
until a horizontal domino-shaped hole at the bottom right corner,
if filled, would form a 2x2 square
Therefore the top left corner cannot be covered,
which negates the assumption
and proves that a 32-domino solution is impossible.
Further reading, courtesy of @Fimpellizieri's comment:
Conway's Tiling Groups PDF
[and height functions]
– W. P. Thurston
Tiling with Polyominoes and Combinatorial Group Theory PDF
– J. H. Conway & J. C. Lagarias
Domino Tilings of the Torus PDF
[and the plane]
– F. de Souza Lima Impellizieri