When counting a given letter in sentences, why do people tend to omit occurrences in certain common words?
What explains the tendency of people to not count the letter F's in the word "OF"?

One possibility to explain this effect is related to the phenomenon of word skipping. When we read, we do not fixate every word, but skip a certain proportion of words while making educated guesses instead. Whether or not a certain word is skipped seems to depend on its length and its predictability, that is, how likely a word is given its predecessor (e.g, Rayner et al., 2011). For your example, the preposition "of" might be very likely to be skipped because it is very short and some grammatical clauses always ask for this preposition. 

Please see PubMed for a more comprehensive literature search on word skipping: http://www.ncbi.nlm.nih.gov/pubmed?term=%22word%20skipping%22

References
Rayner K, Slattery TJ, Drieghe D, Liversedge SP. (2011). Eye movements and word skipping during reading: effects of word length and predictability. J Exp Psychol Hum Percept Perform. 37(2):514-28