Even mutual-fund managers who have an incentive to speak favorably about their corner of the financial markets are bristling at the lofty prices being paid for tax-exempt debt.

An influx of cash into the municipal-bond market—which analysts attribute to investors seeking relief from higher tax bills—has driven some yields to record-setting lows relative to Treasuries after state and local-government securities had their strongest start to a year since 2014. That’s left some waiting for a retreat or finding creative ways...