Don’t ignore them
I assume you’re writing in American English, but you might still encounter accented characters in foreign words. Foreign words arise in two situations:
In proper names, like people and places (Albrecht Dürer, François Truffaut, Plácido Domingo). In names, accented characters must always appear accurately. Otherwise, the name is misspelled.
In loanwords used in American English. Some of these words have become citizens and should be spelled without accents (naive for naïve, melee for mêlée, coupe for coupé). Others have not and should not (cause célèbre, piña colada, Götterdämmerung). Check a dictionary or usage guide.