Signing an affidavit of support as a joint-sponsor makes you liable for 10 years on until they become U.S. citizens only if the foreign national, when they become lawful permanent residents (Green Card holders) applies for government assistance such as welfare, etc. Although historically these obligations are not readily enforced by the U.S. government on a regular basis, it is very difficult to predicate how they will be in the future. So your boyfriend could be held jointly liable by the U.S. government if the foreign national he promised to support goes on some sort of government assistance program.
Not a good idea usually if it is not a close relative, the immigrant has very little incentive to protect the sponsor from potential liability and nobody can predict how immigration law will be enforced in the future.
I agree with the previous answer that the obligations of the affidavit of support largely revolve around public assistance. The affidavit of support would not make your boyfriend liable for other unpaid bills or debts. You can read the exact obligations on the actual affidavit you are signing. You can find a copy of that affidavit at the link below.
A further issue to take into consideration is that a co-sponsorship on this application could affect your boyfriend's ability to sponsor his own family members in the future. If your boyfriend intends to or needs to sponsor another immigrant in the future, he will have to include this petition in the calculations for the poverty guidelines. In other words, if your boyfriend needed to sponsor his parents he would have to meet the level for himself, his parents, and this petition. If he never intends to sponsor anyone else on an immigration petition then this isn't an issue for him.
This answer is not intended to create an attorney-client relationship and is provided solely for informational purposes not as a substitute for legal advice.
From the Form I-864, affidavit of support, that is signed by the joint sponsor: "If you do not provide sufficient support to the person who becomes a permanent resident based on the Form I-864 that you signed, that person may sue you for this support."
Now, some attorneys may argue that the I-864 is somehow not enforceable by the immigrant. However, you and your boyfriend should read the Form I-864 to see what he is agreeing to. The form is free at www.uscis.gov .
It appears to me that there are several places on the form that clearly state that the immigrant can sue the sponsor for support. Moreover, there are several statements on the form showing the sponsor (referred to as I) "have read and I understand each of the obligations".
My experience is that state courts are allowing immigrants to sue based on the affidavits of support. Your boyfriend can search the internet to find those cases.