Is it unprofessional to have emojis in your CTA button?

I don't think it would be unprofessional in every circumstance. It really comes down to knowing who will be interacting with your CTA, how it fits with the brand behind it, and the perception of the industry that the CTA is operating in.

I tend to judge the use of emojis based on age of demographic and the scale of traditional to non-traditional industry and err on the side of caution.

You will most likely not lose a lot by not using emojis, even with an audience that likes them, but could potentially lose by using them with and audience that does not like them.