Well if you had an infinitely thin ring of copper and heated
it, it could only increase in circumference, so the hole
would actually get bigger as the radius of the ring increased.
I believe that your statement that it expands in every
direction is more or less true. What has to be considered,
though, is the real world behavior of such an object where
every atom is expanding just a little bit. The expansion
will create a somewhat complicated stress/strain response
in the metal, and the particular amount of displacement
in a given area would be the result of lessening the overall
stress energy of the metal.
Though expanding from metal into free air is a pretty low
energy thing to do since there wouldn't be back-pressure
from the air in the hole, you must also consider that
there would be a compression of the metal encircling the
air-hole since more metal would have to be squeezed into
a smaller space circumferentially too.
I think that mainly it'd follow the path of expanding
outward radially since that'd seem to lessen strees on the
whole of the piece.
Certainly if it became really hot to liquefy, though, you'd
expect that it'd start to slump/flow/creep to fill in a small
hole.
It'd make an interesting FEM analysis.
Empirically I've never seen a hot 'nut' or 'washer' become
more difficult to put around a cold bolt, or a hot jar lid
become harder to screw around a cold jar. That would
tend to support the theory that mostly it'd just expand
outward as the lowest energy lowest stress option.