There are three common types of mounts.
One is the double-headed plastic spring clip. It has two "teeth" that hook the board when pushed through the hole. The tendency of plastic to return to its original shape is what holds the teeth tightly in the hole.
It doesn't work to remove this clip by just pushing in one side, because the other side still grips the hole. To remove these it is best to use a needle-nosed pliers to squeeze both sides together, and then move it through the hole while still squeezing the pliers. In order to do any of this, you'll have to completely disassemble the computer to access the underside of the motherboard.
The other common style is the "spreader" clip. This is similar to the spring clip but it has two or more teeth and it is not spring-loaded. Instead you slip the tooth-head through the board hole, and then insert a plastic peg into the center of the head. This peg spreads the teeth and holds the board simply because the peg prevents the teeth from moving.
To remove it, you first remove that center peg. Then the toothed head should come out pretty easily. (The peg may itself have an enlarged tip, making it difficult to remove once inserted in the hole.) This type of clip can be removed without getting at the bottom of the motherboard.
Do not try pulling out the plastic mounts by brute force. Especially for this second type, it is quite possible that the printed-circuit board may give first and crack around the hole, destroying any wire traces around that hole.
The third common type of mount is the rubber plug. This is simply squashed through the hole, and the rubber reverts to its original shape when through. This one you can remove by pulling, but be gentle you may tear it off in the hole and need a pliers to get the end out.
The fourth possibility is that the heatsink is epoxied onto the top of the chip. In this case you may very well be stuck and unable to remove it.
If this is the case, you may be able to jury-rig a small 1" box fan onto the existing heatsink. It may be possible to remove any existing fans (or even just leave them there), and run "bread-bag" wire twist-ties through the existing mount holes, then running twist-ties through the new fan mount holes. This will not give a perfect fit, but it may work well enough. Be sure the twisted wire ends are short and do not touch anything. (I had to do this for a motherboard recently..)
