Claim (which I thought would be true, and it was eventually proved in this linked question).
If $B$ is positive semi-definite then there is $z\ge0$ with $z\neq0$ such that
$Bz\ge0$.
(Here $z\ge0$ means that every component of $z$ is non-negative.
By $z\neq0$ we mean that at least one component of $z$ is non-zero.)
(Two proofs of this claim were given by user @daw in my linked question. The same user posted an answer to OP question here too.)
Using the above claim we show that if the set
$S=\{(x,y)|Ax+By\ge c,x\ge0,y\ge0\}$ is non-empty, then it is unbounded.
Take any $(x,y)\in S$. Let $z$ be as in the claim.
Then $(x,y+\lambda z)\in S$
for all $\lambda>0$, proving that $S$ is unbounded.
(Indeed, clearly $Ax+B(y+\lambda z)=Ax+By+\lambda Bz\ge c+\lambda0=c$.)