It’s ultimately up to you, but we’d generally recommend that you honor the request. The candidate may have a good reason for asking you not to contact their current employer. They may fear retaliation if their manager gets wind of their interest in leaving, or they may have a toxic boss who wouldn’t give an accurate assessment of their ability to do the job.
If you wish to contact the current employer despite the candidate’s request, inform the candidate of your decision beforehand and explain what questions you will be asking their employer (dates of employment, job title, etc.). The candidate may opt to let you go ahead with the reference check or they may decide to withdraw themselves from consideration instead of having you reach out. It’s best that they be the one to make this call, especially if contacting their current employer could have negative repercussions for them.