Quick Summary:
This is a bit of a controversial topic. No, you don't have to. Plenty of people have learned to speak the language, yet don't know how to read (or write) the script.
A lot of teachers, however, strongly believe that students should start by learning the alphabet. You can see why: It is how Persian is written, and not knowing it will limit you to auditory materials. There is precious little content out there using romanizations. Dictionaries often have them, and you will be able to rely on those when learning new words and simple sentences, but you'll be hard-pressed to find even a blog post written in Roman script. Imagine not knowing how to read and write in English! So the thinking goes: if you are doing something, do it right from the start. Or as a well-known coursebook says:
[using transliterations] interferes with and ultimately delays the learner's connection with the Persian script and writing system.—Farsi Shirin Ast
We don't really agree with a dogmatic approach though. Learning a language requires a significant investment of time, so nothing is more important than that you keep your motivation. Mindful of that, we would suggest:
1. If you want to mount a serious effort to learn the language, and are confident that you have enough intrinsic motivation to keep going (or maybe you are in a class you can't drop), sure, go ahead and spend some time with the script. It's not really hard at all. At 32 letters, it's isn't any more difficult than learning your first 32 words. You'll have them memorized within a couple of hours. You'll be reading - extremely slowly. The real effort is in reading enough to be able to do so fast and fluently. As you continue learning the language, keep practicing the alphabet. Feel free to rely on romanizations when you become frustrated, but be aware that whenever you are using them, this is time you are not spending improving your reading.
2. If you are less sure about how far you are going to take your studies, or your first priority is being able to talk to people, we don't see why you should make it difficult for yourself. Start practicing speaking! You can always learn the script later, and in some ways you'll have an easier time already being familiar with the vocabulary.