You should contact your insurance company to find out if they will cover the guardrail damages under your property damage liability portion of your policy. State and local laws may differ but we have indeed heard of an individual being billed for damages they did to guardrails, poles, hydrants, walls, etc.
From our research we found that most states' Department of Transportation will bill motorists for damage to guardrails, based on general negligence laws. For the vast majority of motorists, the bill can be sent onto your insurance company, and they will take care of it as part of your accident claim.
While taxes may pay for these guardrails, etc when they are first placed there or for upgrades to them, the DOT usually will go after an at-fault party who damages these objects in order to save the tax money for other items that the DOT has planned in their budget as improvements to our roadways.