Learn Dutch: Het oude vlees wordt taai. - The old meat is becoming tough.

In Dutch, the simple present tense wordt translates to both "becomes" and the continuous "is becoming" in English. Dutch rarely uses a continuous form for a simple change of state. Here, worden is the main verb showing that the meat is changing.

How would I change this sentence to the perfect tense to say "The old meat has become tough"?

You would say Het oude vlees is taai geworden. When worden means "to become", it forms its perfect tense using the auxiliary verb zijn (which becomes is here) and the past participle geworden.

Is wordt being used to form the passive voice here?

No. While worden is often used as a helping verb for passive sentences (like "The meat is being cut"), here it is the main verb meaning "to become". You can tell because there is no past participle or other verb in the sentence.

Why does oude get an -e added to it, but taai doesn't?

Oude comes directly before the noun vlees. Because vlees is a het-word and has the definite article het in front of it, the adjective gets an -e. However, taai comes after the verb to describe the subject. Adjectives in this position never get an -e added to them.