Can you get sad by sniffing onions, just like you can get happy by forcing yourself to smile? I've read about a study where they found that people who were forced to keep a smile-like face were reportedly happier than those who were forced to stay in a frown-like a face. Similarly, could onions induce sadness by forcing one to cry?
I assume you're referring to the experiment by Strack, Martin, and Stepper (1988) in which people rated a cartoon as funnier when they had to hold their face in a smile shape by gripping a pencil in their teeth.

This has been applied to negative affect by Larsen, Kasimatis, and Frey (1992) who had participants furrow their brows during an activity. Golf tees were attached to their eyebrows such that furrowing the brow caused the tips of the tees to touch together. Participants showed more negative emotion in the brow furrowing condition (in which they were instructed to keep the tips of the tees touching).

Both studies, however, hypothesize that the effects are due to facial expression, specifically, to sensations of activating the muscles associated with positive or negative emotions. In the onion paradigm that you propose, it is unclear which (if any) emotion-relevant muscle activations would happen in conjunction with eye irritation.

Nobody that I know of has studied whether eye irritation or production of tears (independent of the accompanying facial expressions associated with sadness) will lead to an increase in sadness.

A competing theory might be that since crying can be cathartic, tear production might make people feel better rather than worse. However the research support for crying and catharsis is mixed and depends heavily on the context of the crying (Bylsma, Vingerhoets, &amp; Rottenberg, 2008).

References
Strack et al., J Personality Social Psych (1998); 54(5): 768-7

Larsen et al., Cognition and emotion (1992); 6(5): 321-338

Bylsma et al., J Social and Clinical Psych (2008), 27(10): 1165-1187