Is the tree in your front garden or is it on the public path?
If on the public path, you'd need to contact your local council. However, unless the tree is rotten / a danger to the public, it is unlikely that they would cut it down just because it is playing host to insects (BTW, having a wildlife friendly tree near you is a good thing).
Spiders and insects have as much right to be here as you do! Besides, there are probably more spiders and insects inside your house than in the tree, you just don't see them...