Unnecessary Text Abbreviations
ECMAScript 6 + HTML (117 123 144)
Using HTML to store the dictionary. Try it online (with any ECMAScript 6 browser such as Firefox).
d=eval(d.innerHTML),t=r=prompt(l=140);for(k in d)r=r.replace(RegExp(k,'g'),d[k])
alert(t.length&gt;l?'Tl;dr':r.length&gt;l)

I'm just blatantly ignoring the currently open questions to this challenge, hoping it won't render my solution incorrect. YOLO #abbreviations
Explanation
The dictionary has been filtered according to the rules (ignoring abbreviations starting with non-alphanumeric character, ignoring multi word abbreviations, only considering uppercase abbreviations, using first replacement if there are options).
To save bytes in the code, the dictionary has also been transformed such that the key is already a regular expression with word boundary match to deal with punctuation.
Then the actual code becomes a matter of reading the tweet, doing all the replacements and checking for the length.
The script used for preparation can be found here. It is based upon the JSON file I have extracted from the website.

Edit #1: Thanks to Optimizer for golfing it down for a win of 21 bytes
Edit #2: Another 6 bytes thanks to Optimizer's tip of using eval.