How can you incorporate light-hearted comedy into your apps? Any good examples of starts that do this in their UI?
The only one I've seen do this is MailChimp. Do usually do this in their error messages by having funny copywriting and sometimes memes.
Apps Copywriting Differentiation
MailChimp's a rare one. Being funny at all is hard to begin with, even more so when it's only written, and even more so to a large, broad group of people. It can be exclusive too; references / memes people don't get just create confusion, and implicitly tell people that your service is not for them. MailChimp uses a light, upbeat tone much more than they use outright humor.
Additionally, the reward is not worth the effort for most startups. Email marketing tools are pretty interchangeable from a features standpoint, so MailChimp's brand is more important for conversion and retention than it would be if it were a different type of service.
For most startups, you're better off differentiating with value than you are with personality. But if you already have that covered or are in a commoditized market like MailChimp, some other examples to check out are: