There is strong evidence (but not a mathematical proof) that the algorithm achieves 7/8 of optimal even on unsatisfiable MAX-3SAT instances.
It can be derandomized using, e.g., the techniques from [2] to yield a deterministic polynomial-time algorithm with the same approximation guarantees.
For the related MAX-E3SAT problem, in which all clauses in the input 3SAT formula are guaranteed to have exactly three literals, the simple randomized approximation algorithm which assigns a truth value to each variable independently and uniformly at random satisfies 7/8 of all clauses in expectation, irrespective of whether the original formula is satisfiable.
Further, this simple algorithm can also be easily derandomized using the method of conditional expectations.
[1] Building upon previous work on the PCP theorem, Johan Håstad showed that, assuming P ≠ NP, no polynomial-time algorithm for MAX 3SAT can achieve a performance ratio exceeding 7/8, even when restricted to satisfiable instances of the problem in which each clause contains exactly three literals.