Do any insects fart or belch?

Yes, they do. At least some arthropods have symbiotic microorganisms that live in their guts and are essential for breaking down food compounds, especially those from plant matter. Among the by-products of these chemical transformations there is methane, the infamous swamp gas... and it goes out the body of the arthropod as 'farts'.

However, it's interesting to know that not all terrestrial arthropods emit methane.

Check this research article from 1994.

http://www.pnas.org/cgi/reprint/91/12/5441

The authors surveyed a wide variety of arthropods and only found methane emmissions in Diplopoda (millipedes),

Blattaria (cockroaches), Isoptera (termites), and Cetonidae (flower beetles). It's good to know that we can't "blame the bugs" in general for suspicious smells!

fart