The only problems I face are with US-issued credit cards. Bank of America has my Israeli address on record and gives me no problem. Amex has my Israeli address and is willing to keep renewing the ccs that I have but will not issue me a new one. TDAmeritrade allowed me to use my Israeli address but, as of a few months ago, required a US mailing address for my debit card only.
My Israeli driving license has never given me a problem in the States and I am allowed to vote in federal elections from Pennsylvania (where I last lived and was registered).
Nowhere do I claim to be an American resident nor would I as it could cause me problems with the IRS for money I earn overseas.