Does avoiding medication that alleviates symptoms shorten the length of a cold?
It is plausible but by no means established that antipyretics (fever suppressors) in particular could increase the duration of infection/symptoms, because fever is part of a functional immune response. 

From Graham et al 1990 (a small [n=56] randomized trial of the use of antipyretic pain relievers in volunteers experimentally infected with rhinovirus):


  Use of aspirin and acetaminophen was associated with suppression of serum neutralizing antibody response (P less than .05 vs. placebo) and increased nasal symptoms and signs (P less than .05 vs. placebo) ... There were no significant differences in viral shedding among the four groups, but a trend toward longer duration of virus shedding was observed in the aspirin and acetaminophen groups.


In other words, it looked like the people who took aspirin or acetaminophen had weaker immune responses and more cold symptoms. They might have been infectious for slightly longer, but it's hard to tell (also, duration of infectiousness is not the same as the duration of symptoms ...)

However, a review (Kim et al 2013) concluded that NSAIDs (non-steroidal anti-inflammatory drugs) slightly (but non-significantly) reduced the duration of illness.


  In a pooled analysis, NSAIDs did not significantly reduce the total symptom score (SMD -0.40, 95% CI -1.03 to 0.24, three studies, random-effects model), or duration of colds ([mean difference] -0.23 [days], 95% CI -1.75 to 1.29, two studies, random-effects model) [emphasis added]


Here is the figure from that analysis:



"Common cold" is often considered synonymous with rhinovirus infection, but may also include mild influenza infections. An observational study on influenza A and two other infections (Graham et al 2000) found


  There was a striking correlation between antipyretic therapy and duration of illness in subjects infected with influenza A and S. sonnei, but not R. rickettsii ... Multivariate analysis suggested that antipyretic therapy prolonged illness in subjects infected with influenza A, but its use was the result of prolonged illness in those infected with S. sonnei.


In other words, to the extent that one can really distinguish causality in an observational study, it seemed that taking antipyretics made influenza illness longer, but that causality went in the other direction for S. sonnei (i.e., people took more antipyretics when they were sick for longer).

The usual caveats associated with small studies and researcher degrees of freedom apply when interpreting the results of these studies ...




J Infect Dis. 1990 Dec;162(6):1277-82 
Adverse effects of aspirin, acetaminophen, and ibuprofen on immune function, viral shedding, and clinical status in rhinovirus-infected volunteers.
Pharmacotherapy 2000 (20), 1417–1422.
(doi:10.1592/phco.20.19.1417.34865)  Effect of antipyretic
therapy on the duration of illness in experimental
influenza A, Shigella sonnei, and Rickettsia rickettsii
infections.
Cochrane Database Syst Rev. 2013 Jun 4;(6):CD006362. doi: 10.1002/14651858.CD006362.pub3.
Non-steroidal anti-inflammatory drugs for the common cold. Kim SY, Chang YJ, Cho HM, Hwang YW, Moon YS.