If $\succsim$ is transitive but irreflexive, then it is asymmetric, proof
A relation that is not asymmetric need not be symmetric. But if $\succsim$ is not asymmetric, there must be elements $y$ and $x$ such that both $x\succsim y$ and $y\succsim x$ hold and these elements are all you need to arrive at a contradiction. 

Also, $x\succsim y\implies y\succsim x$ can hold for an asymmetric relation, namely when $x\succsim y$ does not hold. 

I'd like also to point out that "any" is ambiguous here and might be read as "all" instead
of the intended "some."