Levothyroxine - Quicker Absorption Under Tongue?

The answer for the drug that you are taking is No.

Sublingual use of thyroid medications is always related to natural desiccated thyroid (NDT) drugs like Armour, Nature-Throid or Erfa thyroid. That is because these drugs include the thyroid hormone T3 -- which is faster-acting and has a far shorter-half life than the synthetic T4 found in levothyroxine drugs like Synthroid, Levoxyl, and Eltroxin. Some people feel the effects of T3 very quickly, even within an hour or two, but a levothyroxine drug typically takes much longer to be converted into the T3 that is ultimately usable by the body. Also, an argument that is made for taking thyroid medication sublingually is that by doing so, the thyroid medication is absorbed through the mucous membranes in the mouth, and goes straight to the bloodstream, which prevents the medication absorption from being affected by things like calcium supplements, iron supplements, food, coffee, or fiber. Unfortunately, the active ingredients in thyroid medication have molecules that are so large that it is difficult for them to pass through the mucous membranes. Most of the dissolved/crushed medication ends up swallowed, and moves through the digestive system in the usual way. Other factors also affect sublingual absorption, including oral pH, and salivary enzymes. Hope this helps.

It might be absorbed faster that way, but that isn't the way its meant to be taken. It is supposed to be swallowed. Remember the faster something absorbs the faster its gone from your body too.