I'm on H1B status with one company (parttime) and contracting work with another agency. I did not give the agency my SSN but at end of year, they issue me a 1099. Do I have to file tax return on the 1099 and will that affect my H1B status and future green card application?

Is the contracted work done through the H-1B petitioner?

No. The contracting work is not an H1B petitioner. They do not have my SSN and the SSN is not listed on 1099 Misc.

Then this is unauthorized employment. You should discontinue the second employment immediately. You still need to file taxes as you were paid and there is a 1099 sent in from the company.