Proof complexity and lower bounds
The statement is false for any polynomial time recognizable family of tautologies: the proof system will simply check if the formula is one of them and accept if it is. Proof length of them will be O(1). So I don't think any explicit example is known.

Recall that we don't even know if SAT $\notin$ DTime(O(n)) so we also don't know if SAT$\notin$ coNTime(O(n)) which is equivalent to your question TAUT $\notin$ NTime(O(n)) and follows from a positive answer to your question.

On the other hand, if NP is not equal to coNP, then the family consisting of all tautologies has super polynomial length in any efficient proof system.

Part of what people try to do in prof complexity is to rule out interesting classes of algorithms. A lower bound in a proof system implies a lower bound for all (co-nondeterministic) algorithms whose correctness can be efficiently proven in the proof system (if we can formalize and prove the correctness of an algorithm for SAT in a proof system we can consider its failing execution to find a satisfying assignment as proofs of the tautologihood).