If insurance pays for repairs after a fire at my rental property, is the repair cost still a current expense?

Yes. Whether repairs after a fire are current or capital in nature is judged the same way as any other repair, by looking at whether the work restores the property to its original condition or improves it beyond that, and the fact that insurance is paying for some or all of the cost doesn't change that underlying character at all. Insurance proceeds simply offset the cost of the work; they don't transform a repair into something else.

So if the fire damage is genuinely repaired back to roughly what existed before, the repair costs are current expenses in the usual way, fully deductible in the year incurred, regardless of how much of that cost you personally paid out of pocket versus how much insurance covered. Where it can shift is if you use the opportunity to upgrade beyond what was there before, say installing significantly better materials or expanding the space during the rebuild, since that portion would be analyzed as a capital improvement under the normal repair-versus-improvement test, exactly as it would be without any insurance involved. Keeping the insurance settlement details, the contractor's invoice, and a clear description of what was actually rebuilt is useful if any portion of the work looks like it goes beyond a straight repair.