The correct answer is "is." Let's break the sentence down into it's parts of speech.
The furniture in our classroom ____ uncomfortable.
"The furniture...uncomfortable" = nominative
"in our classroom" = ablative
"is" = third-person singular present active indicative
"are" = third-person plural present active indicative
In this sentence, the verb "to be" is supposed to equate the subject to the adjective ("uncomfortable") that follows it. The verb has to match the number/plurality of the subject. The subject of a sentence is also known as the nominative. As indicated above, "furniture" and its articles and adjectives are all in the nominative. Thus, "to be" has to match "furniture." "Furniture" is actually a weird word in English because it is considered to be uncountable. This means it only has a singular form and does not ever have a plural form (e.g. the plural of furniture is furniture). So, we would choose "is" because there is no plural form of "furniture," even if it was supposed to be plural.
There actually once was a plural of furniture, furnitures. It fell out of use during the 1900s, and would have taken the verb "are."