Latch Issues
A fence gate that won't close may be an indicator of an issue with the gate latch. Latches that rust because of exposure to the elements may fail to close properly, leaving the gate susceptible to blowing open in a breeze. A broken latch can cause the same problem and may need replacing before the gate will shut and remain closed. To keep your fence gate latch in good condition, oil it regularly and check to ensure that the mounting screws are tight on both the fence and the gate itself.
Obstructions
Another problem that will cause a fence gate not to close is an obstruction that prevents the gate from swinging shut so the latch can engage. Even if the gate comes within a fraction of an inch of closing fully, the latch might not catch. This can occur if stones or wood chips from your landscaping come loose and serve as doorstops for the gate. Grass and other plants can also grow up around the gate posts, restricting its movement.
Security
Whatever the cause, a fence gate that won't close presents a vulnerability in terms of security. Whether your fence is primarily installed for privacy or to keep people off your property, an open gate is a poor deterrent. You can close and lock your gate temporarily with a padlock and length of metal chain. Self-closing gates use spring mechanisms to shut automatically, which might be a useful feature if security is an issue for you.
Lack of Barriers
Even if security isn't a primary concern, a gate that won't close represents an incomplete barrier for your yard. This can be a problem if you rely on your fence to keep pets or children from leaving the yard and finding their way into the street or onto a neighboring property. If your gate won't close and you can't lock it closed temporarily, you may need to use plywood or another large, solid piece of material to erect a temporary barrier.