Translation:That is not my husband.
You are right. 夫 is fū, 1st tone.
A Chinese character may have more than one tone, but for each of the meanings the character is applied, there is a fixed tone.
And then there are some rules that requires us to change the tone in the case of certain sequences of the tones. However, None of these rules would changes other tones to the 3rd tone; It can only occur in the other way.
Thus the tone affects delivery and perception of meaning during verbal communication. It is certainly true that different regional accents using non-standard tones exist. Nonetheless a learner of Chinese may find it very confusing when exposed to pronunciations inconsistent with the standard one at a premature stage.