Montressor needs to make sure that his plan goes off without a hitch. If Montressor were to act indifferent to Fortunato's health, Fortunato may suspect that Montressor was up to no good. Montressor is simply trying to put Fortunato at ease, and play the concerned friend who would never do anything to harm his "friend" Fortunato--see, Montressor is even concerned for Fortunato's health. There's no reason Fortunato shouldn't follow this man deeper and deeper into the underground.
It also serves as a delicious irony for Montressor. He can act concerned about Fortunato's health, even though Montressor knows that Fortunato's cough and general health are the least of his concerns right now and will not be the cause of Fortunato's death.
Many elements of Poe's story serve a double purpose for reasons of economy. Poe's main reason for giving his character Fortunato a bad cold was to make it difficult for him to talk. Here is the most significant dialogue pertaining to Fortunato's cold:
"How long have you had that cough!"
"Ugh! ugh! ugh! -- ugh! ugh! ugh! --ugh! ugh! ugh! --ugh! ugh! ugh! -- ugh! ugh! ugh!
My poor friend found it impossible to reply for many minutes.
The bad cold of the intended victim makes his motivation to sample the Amontillado appear even stronger. It gives Montresor excuses to urge him to turn back (thereby showing his friendship and lack of any sinister motive). It causes Fortunato to drink more wine, thereby becoming drunker and easier to beguile. But most importantly it prevents Fortunato from asking a lot of awkward questions. Poe specified in his famous review of Hawthorne's Twice-Told Tales that not a single word should be included in a short story which did not contribute to the single effect; yet he uses fifteen "ughs!" This means the cold is very important.
If Fortunato hadn't been hampered by his cough he would almost certainly have asked such questions as:
When did you buy this Amontillado?
Whom did you buy it from?
Who was the exporter?
How much did you pay for it?
Who was the dealer?
Why did you store the cask so far from the bottom of the staircase?
Where are you taking me?!!
Poe avoids all such questions by the simple expedient of giving Fortunato a bad cold. Evidently Fortunato knows more about Amontillado than Montresor, so Montresor would have had a hard time answering questions about his non-existent cask of Amontillado, and Fortunato's suspicions would have been aroused.
Montresor appears concerned about Fortunato's health to cover up his evil intentions of murdering him. Fortunato feels flattered that Montressor is so concerned about his cold and instantly walks into the trap laid for him:
"My friend, no. It is not the engagement, but the severe cold with which I perceive you are afflicted . The vaults are insufferably damp. They are encrusted with nitre."
"Let us go, nevertheless. The cold is merely nothing."
Montresor then gets Fortunato drunk by telling him that the wine will keep them warm. Montresor tells Fortunato, "a draught of this Medoc will defend us from the damps." Medoc is a red wine from the SW of France.
The burial place of the Montresor family-the catacombs-is damp and very cold and so Montresor offers Fortunato a bottle of wine to drink so that Fortunato will not feel the cold.
The real reason Montresor gets Fortunato drunk, however, is to create a false sense of bonhomie so that Fortunato will not suspect his evil intentions of murdering him:both of them even exchange toasts. Further, by getting him drunk Montresor slows down Fortunato's reflexes so that he will not be able to escape.