1. Provide technical support to customers by email and phone.
2. Set up environment to reproduce critical customers issues inhouse for developers or quality engineers (QEs). Make sure that issues passed on to QEs are genuine issues.
3. Be an active part of the Quality Assurance process - track bugs,
test fixes, write technical notes/documentation, attend/provide training etc.
4. There are several roles for support engineers - general support, premium support (assigned to one customer only for all their issues), advanecd support (support only for the most advanced features of a product).
5. At times, get involved in POC (Proof of Concept) cycle. This is a pre-sale demo to the customer showing how well the product fits their needs. This involves on-site travel, generally.
That's what Tech Support Engineers do in my company.
Hope that helps!
Amit