I just wanted to chime in with one answer which the others here have already stated but without specifically mentioning this particular aspect... Nipples.
Don't giggle.
I know you're giggling.
Actually, nipple burn is a serious annoyance for anyone doing gi work.
And I'm talking about guys.
I assume women have some kind of athletic bra and t-shirt underneath, so they're protected, more or less.
If you've ever worn a grappling gi (Judo / BJJ), then you know those things are like cardboard.
They're thick and heavy.
And when you have an opponent who constantly grabs your gi, that thing will slide up and down, side to side.
The effect is that your nipples will feel like someone took a cheese grater to them.
I used to train in judo in high school, and we didn't use rash guards under our gi tops.
I wouldn't even realize that my nipples were getting scraped by my gi until I went home and hit the shower.
Once the water hits, it's like some kind of bizarre form of torture.
Either that or you can think of it as a kind of masochistic badge of honor.
If they don't permit rash guards, I wonder if they allow you to tape your nipples in IBJFF competition?
Or maybe you could claim to be transgendered, and they'll allow you to wear an athletic bra underneath.
Hehe.
Just kidding.
martialarts.stackexchange.com/users/1555/Steve Weigand
Pros Prevents chest hair falling off into partner's mouths.
Lessens the chance of skin infections.
Reduces incidence of gi burn and mat burn.
(These are wimpy reasons.)
Increases modesty, for instance, hiding fat and reducing skin-to-skin contact that makes some people feel weird.
Warmer for winter months.
Increases friction when the gi is not between you and your partner, for instance, increasing control when chest-to-chest in side control.
Cons
More laundry.
Prevents display of muscular physique.
Notifies people that you're a big wimp about pinches and scrapes.
Hotter in summer months.
Increases friction, making some escapes more difficult.
This can also give your opponent control over you, for instance, giving them a tackier surface to play guard against.
martialarts.stackexchange.com/users/347/Dave Liepmann
I and some of my friends wear rashguards or even just T-shirts under our Gi's during class simply to lessen the amount of exposed skin and the amount of skin-to-skin contact.
So far as I know, there are no other advantages.
As for the drawback, I am not aware of any other than having a rashguard that will need to be washed along with the GI after class.
If it is a looser T-shirt, I have seen hands caught in its collar when going for a gi collar grab, but I have yet to see that provide either a meaningful advantage or a disadvantage in the actual round and a tight-fitting rashguard should not have that issue.
I believe that the reason they are banned in gi-competitions is simply tradition and it has not been strictly enforced in the competitions I have attended.
martialarts.stackexchange.com/users/1253/TimothyAWiseman
I wear a rash guard basically to avoid Skin to Skin contact.
Skin to mat contact.
Sweat contact.
Unintentional body hair ripping.
Since my dojo also have different MA sessions sharing the mat at different times, I am abit cautious about the mat.
I have heard tons of stories from the web about viruses contracted from dirty mats.
martialarts.stackexchange.com/users/1650/nigelhanzo
Although not your question directly, you did mention in your post
"
Rashguard are forbidden in IBJJF Gi competition.
"
IBJJF bans rashguards because they get in the way while doing collar chokes.
Oftentimes the choker will get a hand full of rashguard when trying to attack the opponent's neck, which makes the choke harder to finish.
martialarts.stackexchange.com/users/8917/C Henry
I wore a heavy gi doing Hapkido 3 years and judo another 2, I never noticed any nipple problem.
That is until I started distance running last year, my sweaty shirt rubbed them raw, so I taped them - doing that for judo practice could be dangerous to chest hair during randori.
one difference between then and now (20 year gap) is, where I was about 160 lbs, now I am 180 and have developed some man boobs :(
maybe that's a factor with regards to the gi.
martialarts.stackexchange.com/users/5498/Haggisbreath
Other people have told you the pros and cons.
All I can add is that they are banned for the same reason other martial arts (or even schools) ban items of clothing: they're not part of the prescribed uniform.
e.g. I almost got disqualified from the 1997 South African Taekwondo championships because I wore sneakers during the breaking event.
Luckily I was in the grandmaster's good books back then, so he just gave me a warning.
martialarts.stackexchange.com/users/1200/Captain Kenpachi
I would add that a well-fitted, cotton t-shirt might be preferable to a rashguard, given that the t-shirt traps less heat, while still offering rash and burn protection.
martialarts.stackexchange.com/users/7401/jacefarm
Rashguard are forbidden in IBJJF Gi competition.  What are the drawback and advantage of wearing a rashguard under the Gi?
What are the advantage of wearing a rashguard under the Gi in BJJ?
martialarts.stackexchange.com
<brazilian-jiu-jitsu>
martialarts.stackexchange.com/questions/4654
martialarts.stackexchange.com/users/412/ucsky
A rashguard lessens the chance of skin infections, and reduces the incidence of gi burn and mat burn. A rash guard generally reduces skin to skin, skin to mat and sweat contact. Drawbacka and advantages of wearing a rashguard under the GI include preventing the ripping of body hair,with  less chance of skin infections, as well as  avoiding skin to skin and sweat contact.
Drawbacka and advantages of wearing a rashguard under the GI include preventing ripping of body hair, less chance of skin infections, and avoids skin to skin and sweat contact.