if G is finite, every element has to have finite order, so for a finite group, Tor(G) = G.So suppose G is finite.
this is really quite simple. suppose a,b are of finite order m and n, respectively.
then because G is abelian
so ab must be of finite order.
for the non-abelian case, the only example that comes to mind is the free group on two generators a,b subject to the relations .
clearly ab is of infinite order.