I have an ally that is exceptionally exhausted. If I polymorph him into another form, do the negative effects of exhaustion—such as reduced speed—apply to the new polymorphed form?
Exhaustion levels are a type of condition, and ...
A condition lasts either until it is countered (the prone condition is countered by standing up, for example) or for a duration specified by the effect that imposed the condition.
(PHB, p. 290).
Nothing in the description of the Polymorph spell indicates that it counters any conditions, nor does it specifically mention removing exhaustion levels. A character who is polymorphed while prone, frightened, or paralyzed thus keeps those conditions, and likewise they would keep any levels of exhaustion they had before they were polymorphed.
Some creatures are immune to some conditions, including exhaustion. If a character with one or more levels of exhaustion were polymorphed into one of these creatures, they would ignore the effects of those exhaustion levels while they were polymorphed. Generally, the creatures that are immune to the exhausted condition are angels, constructs, and incorporeal undead, and thus such a transformation would require the 9th-level True Polymorph spell.
Assuming that you are referring to the Polymorph spell (because there is no general polymorph rules all polymorph effects are specific unto themselves), it says:
The target’s game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains its alignment
"Statistics" are those things that are in the creature's stat block. Conditions like exhaustion or unconsciousness or fear are not part of these so whatever the person has before, they continue to have after the spell takes effect.