Some do, some don't. Even one person's pronunciations can shift depending on the situation.
I pronounce months /mʌnθs/, with the θ. I think most people I know personally also pronounce the θ. But not all. When I hear it without the θ, I cringe (inwardly). To my ear, that sounds sloppy. There are also some East-coast regional accents that make it /mʌnts/. I've even heard /mʌmfs/.
When I was a kid, I pronounced clothes /kloʊz/, without the ð. That's how most Americans pronounce it. But I have heard some people pronounce it /kloʊðz/. To my surprise, many of those people grew up in the same region that I did, central Ohio. When I heard that pronunciation, it sounded clearer, more elegant, and more formal than my pronunciation, but also a bit fussy (to my ear, not theirs). As an adult, my pronunciation of this word tends to vary. While giving a talk at a research conference, I would probably pronounce it /kloʊðz/; while asking about when the washing machine will be ready, probably /kloʊz/.
If you want to sound educated, intelligent, high-class, or formal, then articulate every vowel and consonant clearly, in the standard way. If you want to sound uneducated, not-so-intelligent, low-class, or casual, then slur anything that's hard to pronounce. There's more subtlety to the way people perceive clarity in speech than that, but if you master clear speech first, you'll be fine. Later, you can learn when to "lax up".
By the way, for some time when I was a little kid, I was puzzled about how you could make "clothes" singular. No one says /kloʊ/. And "clothe" (/kloʊð/) is a verb. I had to wait a long time before I learned the phrase "article of clothing", which is the singular.