If there were no quotation marks in the message
Imagine that the punctuation of the posted example had not included any quotation marks:
You must post precautions for usage, etc. where applicable.
In that case, adding a comma after etc. would be consistent with the style advice offered by Chicago Manual of Style, 15th Edition (2003):
6.22 "Etc. and "and so forth." The abbreviation etc. (et cetera, literally "and others of the same kind") is traditionally both preceded and followed by a comma when it is the final item in a series. Such English equivalents as and so forth, and the like, are usually treated the same way. ...
[Example:] Cats, dogs, parrots, etc., in transit must be confined to cages.
But Chicago also endorses an alternative approach:
An alternative and quite acceptable usage is to omit the second comma (after etc.), punctuating such expressions in the same way as the final element in any series.
In short, Chicago notes that "traditionally" a comma would follow etc. in a list where etc. is the last entry in the list—but it also acknowledges that basing the decision about whether to add a comma after etc. on whether you would have added a comma to any other word that happened to be the final element of the series is also "quite acceptable usage."
Bryan Garner, Garner's Modern American Usage (2003) endorses the last part of Chicago's advice, without any shilly-shallying over "traditional" practice:
Punctuating. Punctuate around this phrase ["et cetera" or "etc."] as if the words and others were substituted in its place. For example, don't put a comma after etc. if it's the tail end of a subject <side dishes of carrots, potatoes, broccoli, etc. are also available>.
One odd thing about the "list" in your example is that it consists of a single entry ("usage") followed by "etc." For purposes of defining what etc. (that is, "others of the same kind") might be in this instance, the single term "usage" isn't much to go on. To improve the message's comprehensibility—still supposing that the message lacked quotation marks—you might want to remove the list-implying comma after "usage" and reword the message along these lines:
You must post precautions for usage and other relevant warnings where applicable.
Since there are quotation marks in the message
The foregoing discussion focuses on how to handle a message that lacks quotation marks. In contrast, your example does have quotation marks—and it seems to me that the message within the quotation marks is properly read not as an exceedingly brief list, but as a simple quotation whose content is irrelevant to the question of how to punctuate around it.
With regard to punctuating the quotation, it doesn't matter whether the phrase enclosed in quotation marks is "precautions for usage, etc." or "curb your dog," or "slippery when wet": In any of those instances, the quoted wording is structurally a unitary thing—a quoted warning—and you would no more add a comma after the last word in any of those quotations than you would after the word warnings in this sentence:
You must post warnings where applicable.
My conclusion is that there is no reason to add a comma after etc. in this poster message:
You must post "precautions for usage, etc." where applicable.