Some ideas for climax without combat:
1. Theft/secrecy/smear campaign PCs reputations are damaging them in some way. Perhaps they are TOO loved by the populace and can't get a single night's rest without someone calling for their aid. Perhaps they are a witness protection program scenario. Perhaps lawsuits are coming after them. Perhaps they risk incirmination/condemnation if evidence (true or false) leaks out. Their task is to destroy the evidence, concoct secret identities, or destroy/build up their reputation.
2. Find the Killer: A classic mystery trope is "find the killer." You might want to use some urban tracking rules from Crime & Punishment (Atlast Games). The climax occurs when the PCs finally reach the criminal, only to find they need to negotiate with him because...
- Innocent lives are in danger (He has someone trapped in a coffin and their air is running out, he has a secret weapon hidden in a population center, only he has the power to defeat the demon he summoned, etc)
- Only he knows some information they need
- Back-up is on its way and will be very put out if they kill him (he could be bluffing)
3. Chase: I'd recommend Hot Pursuits for an epic chase that your players will love you for.
4. Moral dilemma: If I kill my enemy, do I end up too much like them? Do I honor my dying foe's last request? Is the greater good more important than my personal welfare? Is it better to die with honor or to cheat and live? These moral dilemmas are usually quickly resolved, making for a very fast climax. They can be lengthened by combining them with other scenarios (escape, chase, etc).
5. Trial: I LOVE running these scenes, though they are more suited for justice-minded PCs who don't kill their enemies. A great example is found in the M&M adventure Church & State. What happen if the bad guy gets off scott free?