There are already answers with simple examples, so why one more? Because it is interesting to look at the general pattern. The wanted property is a kind of symmetry, so we should look for some asymmetrical joint distribution for $X,Y)$. Si if $(X,Y)$ has a permutable distribution in the sense that $(X,Y)$ and $(Y,X)$ have the same distribution, so that for the joint cumulative we have $F(x,y)=F(y,x)$ for all $(x,y)$, then the sought-after property will hold.
Let us use copulas. Let $F$ be the joint cdf (cumulative distribution function) and
C(u,v)= \P(F(x) \le u, F(Y)\le v)
By the Fréchet–Hoeffding copula bounds (see linked wiki article above) we have
$$ W(u,v) \le C(u,v) \le M(u,v) $$
where $W(u,v)= \max(u+v-1,0)$ and $M(u,v)=\min(u,v)$. Both $W,M$ are copulas. $W$ describes the anti-monotonic case $X=U, Y=1-U$ for $U$ some uniform random variable. Now you can check that $W$ gives a counterexample. $M$ corresponds to $X=U, Y=U$ which is not a counterexample.