No.  Policyholders should not fear their insurer dropping them or raising rates if they challenge a payout.  They simply will not drop you because of a claim.  However, they will drop you if you are a risk to them, for example, if they were to find out that you are storing propane tanks in the house or you have exposed wiring.
As for premium rate increases, this is determined by the nature of the claim, not the cost of recovery.