Neither! The sentence is completely wrong. For the 'so' structure to work, the second half of the sentence has to mirror the first. If you are comparing being overweight with smoking and drinking, both have to be the subject of the clause. For example, "I know that being overweight causes health problems, but so do smoking and drinking".

"I know that there are health problems associated with being overweight, but so are smoking and drinking." 'are' is correct since you are referencing two or more things. However, this sentence is odd because you are saying that smoking and drinking are also associated with being overweight. I don't think that is what you mean to say. I think you mean to say that there are health problems associated with smoking and drinking. If that is the case, this sentence is wrong. "I know that there are health problems associated with being overweight, but the same is true of smoking and drinking." " ....smoking, drinking, and the list goes on." I can't say for sure that this is incorrect but it definitely sounds odd to me. I wouldn't say it this way. Maybe, "...smoking, drinking, and a number of other things." ... or something like that. I am interested to know if anyone else feels that this phrasing is awkward.

is

Thank you very much.

I do not know if this will help you regarding the usage of the words "is" and "are'. You use "is" for singular words and "are" for plural words. I think you should use "is" in the way you have expressed the sentence, but I feel your sentence is not correctly written and to include "and the list goes on" I would have said the following. I know there are health problems caused with being overweight, through smoking and drinking plus many other reasons too long to list. I hope this is of help. Best wishes Roger.