What issues could arise in a resume if the improper verb tense was used?

The most obvious thing that could arise is that the people reading the resume might not understand whether some part of your career that you've mentioned is over, or is still going on. The context would probably make it clear, but not always; and that brings us to the second obvious issue, that the people reading your resume might conclude that your grasp of English wasn't strong enough for them to want to hire you.