Your resume should look professional.
What this means may be somewhat domain-specific. For example, some people argue that graphic designers should use graphic design in their resumes (example). This would clearly allow icons (and much more). But this is disputed, with others arguing against (example).
At least for most fields, though, a conservative approach is advised. Don't deviate too far from the standard. Make sure your resume is a clean, clear presentation of the key information.
Overall, my take (as someone who does a fair amount of hiring for research/software development):
I see no problem with including icons as long as they are tasteful and fairly subtle.
Emoticons are inherently unprofessional/informal, but the same doesn't necessarily apply to icons.
They might even help set your resume apart, as part of an overall good design. But make sure they actually contribute to the document. Don't add them just for the sake of doing something "different".
For example, I think something like this could work quite well:
Touque University, BS in headwear science
💼 Work experience
Professional briefcase carrier, 2012 - present
Using unicode icons here, they might not show up for everybody.