ATC can tell you to do anything or change anything they want, yes you
will want to carry a low altitude enroute chart and have it handy.

when I departed ontario, ca. (ONT) last week, it was my intentions to
cancel IFR as soon as I was on top of the fog and was VFR so I did not
pull out my chart. the initial clearance was simple enough, from ONT,
runway heading to 3000, direct PDZ, climb maintain 4000 , direct HND
(Henderson Nevada), after take off, they changed it up on me and told me
after PDZ intercept v448 - HEC, then as filed. so I had to dig out my
chart and find v448 while in a climb, in IMC and about 22 miles from PDZ
where I would be turning onto v448.
No biggie tho if you know what your looking for. After getting
established on v448 I was able to cancel IFR and continue VFR.

Andrew Sarangan wrote:

If you file to a VOR direct, can ATC clear you along an airway
instead, and expect you to look up the airways? In other words, how
important is to carry an enroute chart if you don't plan on using
airways?