This part should be driven with about 10V to get the minimum guaranteed 'on' resistance. The relevant section is below:
At as much as 4V (the line above) the MOSFET may only be conducting 250uA, so
hardly on at all.
"Typically" it will turn on fairly well with a 5V input, but with 3.3V it may conduct very little current. There is no guarantee your particular part is typical, but if you are actually driving it with 5V there may be something else wrong.
+/-30V is the absolute maximum gate to source voltage. You should not get anywhere near that voltage. Above that, destruction can occur, and there is hardly any gain to be had from more than 10V-15V drive.