What happens to the word harmless's base word when suffixes beginning with a consonant are added?

Nothing. The only regular suffixes added to "harm" that begin with consonants are - ful and -less, creating the Words harmful (harmfully) and harmless (harmlessness, harmlessly).

Some Words change their spelling, or double the ending consonant depending on whether the suffix begins with a vowel or consonant, but harm is not one of them.