Can i catch an illness by breathing in someone elses burp?

In theory, yes...: If the burp-er is sick & in burping, microscopically sprays the area around them with their illness-causing bacteria & you are close enough to them to inhale said bacteria, and there are enough bacteria to overpower your natural immune system, then yes, you could "catch an illness by breathing in someone else's burp." But that's a lot of ifs...protect yourself by remembering to wash those hands!