"Like you mean it" is not only used sarcastically. That would not be my first assumption. Kiss me like you mean it. (a song lyric) Play your part like you mean it. Act like you mean it. Basically, it means do something with passion. Be genuine. Don't do it half-heartedly. "like you should" isn't really a standard expression to my knowledge. If it means what it sounds like I'd say instead "like you're supposed to" Clean up after yourself like you're supposed to. Treat everyone with respect like you're supposed to.

'Like you mean it' usually means 'As if you mean it !' (ie I don't believe you). It's a sarcastic response usually. Someone tells you something but you don't believe them. You don't think they are serious or genuine. Examples: (1)Two people have an argument and one says sorry. The other says 'Like you mean it'. The second person is still angry and believes the apology is not genuine. (2)I failed all my exams and tell my friend I'm going to stop partying and study better this time. My friend doesn't believe me and sarcastically replies 'Like you mean it'. 'Like you should' isn't a stand-alone expression I've ever used. I've never heard it either.