Do you recommend CSF analysis in young patients with imaging findings consistent with progressive supranuclear palsy?

This is a difficult question and I'm not experienced enough to give you a great answer. In the end, it depends on the age of onset (age <50 would give me pause), rate of progression, the inclusion of expected PSP symptomology (supranuclear palsy, akinetic-rigid axial parkinsonism, spontaneous retropulsion, and/or frontal dysexecutive cognitive impairment), the exclusion of systemic symptomology (night sweats, fevers, weight loss, known cancers, etc.), levodopa-response (a positive response would argue against secondary causes but an absent response is seen in PSP), and your gestalt.

I could not find great retrospective studies detailing the natural history of PSP but it is assuredly a much more rapid progression to severe Parkinsonism than Parkinson's disease. I have not found much more than a few case reports detailing paraneoplastic PSP phenotypes. Those patients were typically severely impacted within 6 to 12 months, had severe cognitive impairments, and had no levodopa response. 

These are overall difficult cases. No one would fault you for CSF analysis and repeat neuroimaging.