I'm not going to let my neighbor use my grill
Why don't you want him to use your grill?
I feel he might damage it and make it unusable.
In order to maintain good relationship with your neighbors it is good to be polite. You can make up an excuse if you don't think he will treat it right and maintain a good relationship.
I don't see why I should do that. I feel I should make an excuse to make him not come over too.
If you have to live next to someone don't you think you should be honest and try to keep the peace. I think it is a bit silly to think he might destroy your grill. They are really resilient and nearly impossible to break!