Farsi was influenced by Arabic after the Islamic expansions but ultimately it's a completely separate language.
To answer your question, no Arabs cannot understand the least bit of Farsi and vise versa.
EDIT: I'm confused. I don't understand why people are giving my answer thumbs down. The question is "do Arabs understand Persian?" and me being an Arab born and lived in Arabia my answer is from first hand knowledge is no they can't.
If we want to go into further details, here's an extract from the Wikipedia article on Iran, under section Languages:
"In the late 8th century, Persian was highly Arabized and written in a modified Arabic script. This caused a movement supporting the revival of Persian. An important event of this revival was the writing of the Shahname by Ferdowsi (Persian: Epic of Kings), Iran's national epic, which is said to have been written entirely in native Persian. This gave rise to a strong reassertion of Iranian national identity, and is in part credited for the continued existence of Persian as a separate language".
To summarize, Islam entered Persia and a big part of the Islamic tradition (Quran and Hadith) are written in Arabic scripture. While it is NOT mandatory for a non-Arab Muslim to learn Arabic, all Muslims are required to be able to recite some Quran during prayers which consequently results in non-Arab Muslims inevitably learning some form of basic Arabic or in the least be exposed to numerous Arabic words and phrases hat are present in religious recitations/duties.
I'm a very well travelled person and I have met people from pretty much every corner of the earth and almost always I can tell a Muslim from using Arabic words or Arabic-sounding words in their own language. For example, Tatars (a group of people that reside in western Russia/Eastern Europe) have a high percentage of Muslims. When they talk among themselves, they use a form of Russian mixed with the old tatarian language mixed with thousands of Arabic words (which is weird to listen to someone talking Russian mixed with Arabic). The same applies to people from south east Asia (Indonesia, Malaysia, etc) and India and Pakistan.
In short, those civilizations use a number of Arabic words that were assimilated into their culture after Islam was introduced to them but that by no means they can understand Arabic or that Arabs can understand them. Not any more than Spanish speaking people can understand Tagalog (the language spoken in the Phillipnes which was BASED on a mix of Spanish and original Malay languages).
I have to strongly disagree with the person stating that 80% of other indo-Aryan languages are based on Arabic. If anything, it would be the other way around where Arabic is only 20%.
I also have to point that "an Arab person who can learn Farsi in 3 weeks" is nothing short of a genius. Learning a whole new language in 3 weeks? Perhaps your friend can do a favor to humanity and make a language learning course sir? He would be filthy rich if he teaches the rest of us normal people how to learn a WHOLE NEW LANGUAGE in3 weeks ;)
Hope this clarifies it.