It's very rare to need the resistor, the reason for it is to prevent the crystal been over driven - some crystals might need it, but it shouldn't prevent it oscillating at all.
First off you need to make sure it's all connected up correctly, in particular make sure that all Vcc and Vdd connections are made - you have to use them all!.
If you have access to a scope, check if it's oscillating - you MUST use a x10 probe, a x1 will stop the oscillator.
If you have no hardware problems, and presumably the PIC verified OK after programming?, you may have a software problem.
No, crystals aren't polarised, like anything they can be damaged by excessive heat, but with any slight soldering ability at all it wouldn't be a problem.
Put this code into your chip, make sure that port c (or whatever you choose) has bit 0 as an output and bit 1 as an input e.g.
MOVLW B'11111110' ;MASK TO MAKE BIT 0 OUTPUT, REST INPUT
TRIS PORTC ;TRISTATE PORT C
TEST
BSF PORTC,0 ;TURN ON PORTC,0 THIS IS AN LED OUTPUT
BTFSS PORTC,1 ;IS PORT C HELD HIGH?
BCF PORTC,0 ;NO, TURN OFF PORTC,0 THIS IS AN LED OUTPUT
GOTO TEST ;LOOP FOREVER
Ok, IF the pic is running then when PORTC,1 is high (5V) the led will light, when it is low, it will go out...... You may have to change the ports and I/0 lines to suit your design.