Technically, the Arrow television series does not ever tell viewers whether the costume the Arrow wears is armored. Given the nature of his work, battling armed gunmen on a fairly regular basis, there should be some kind of armor resistance woven into the suit. Given this is the DC Universe, complex and capable armor technology should exist. (See: Batman)
We are forced to consider the canon and delve into a tiny bit of historical speculation between the Arrowverse and the canon DC Universe.
In the canon DC Multiverse
In the older versions of the DCU, Pre-Crisis and Post-Crisis, Green Arrow wore only the tiniest bits of armor in strategic places because he needs mobility in his arms and upper body.
During the famed Neil Adams 'Green Lantern/Green Arrow' period he wore almost no armor at all and this was his canon appearance for decades. Since his adventures depended on his ability to find and get the first strike on the enemy, he dressed lightly, in order to put out the incredible volume of arrows he can release.
Green Arrow in his first costume now considered the Golden Age, Earth-Two costume; his most well known appearance during the Silver Age, where he wore almost no armor.
Green Arrow would be considered a strong offensive fighter, putting a volume of his best attack into play before the bad guys can retaliate. Done properly, most hardly get a second chance due to his astounding accuracy. Black Canary once counted he was able to put 39 arrows into the air in under a minute!
Green Arrow was considered to be a very capable hand to hand combatant, which may have been accented by his wearing of light armor; if you have less armor to move, there is a benefit to your agility and overall foot speed. During his Justice League years, he had the additional benefit of fighting at range while more damage resistant heroes provided him with the opportunity to shoot without fear of taking damage; like a sniper protects moving ground troops.
Such mobility meant he was only armored (if he wore armor at all) across his chest/back, calves/shins and possibly knees. He may have also had a bit of armor on his forearms. In his most recent appearances in the DC Universe he is wearing the most apparent armor the character has ever worn.
In the Arrowverse
We are never privy to the creation of the costume in Arrow, so we don't know its specifications, whether it is actually armor or just meant to be camouflage. Given Queen's previous wealth, he should be able to afford the best in personal body armor, making it as light as possible while still offering superior protection.
Given that the Arrow expects to get shot at, and is often being shot at, he would want at least a degree of protection over his vital organs. We do know his current outfit gives him maximum mobility allowing him to dodge and fight hand-to-hand quite well.
Looking at the costume, there doesn't appear to be much in the way of visible protection, beyond the leather exterior. It does not mean he isn't using a form of kevlar/nomex (for protection against small arms or fire) or SAPI ceramic plating (providing limited protection against small arms or arrows) underneath the outer layer of the costume.
Arsenal's impending costume does look very different from the Arrow's but does not have any visible outer plating in the form of visible armor.
For comparison, consider the one of the Arrow's most dangerous foes, Deathstroke.
Deathstroke's armor which has to withstand repeated attacks from bullets, shrapnel and occasionally arrows.
Note the numerous ceramic plates covering vital portions of the armor. Deathstroke's fighting style places him in the center of enemy fire and having visible, dense armor plating makes sense considering his greater than normal human strength and agility.
He also does not need quite the level of mobility to use his firearms or sword in comparison to using a bow.
UPDATE: It would appear a confirmation was made in the cross-over episode The Brave and the Bold (a tribute to the crossover comic by the same name) in which we discover Oliver's costume indeed DID NOT HAVE ANY ARMOR. (Pathetic and a bit crazy...)
- Team Flash's technologist and fabricator Cisco is revealed to have created a new costume with kevlar armor woven into it, and a hood for its "sentimental value".
CISCO: We got a gift for you, too, Oliver. Early Christmas present.
Turns out S.T.A.R. Labs' industrial fabricator was based out of Starling, so Polymer Kevlar weave. It's 25% lighter and can carry 15% more gear. I've got designs I'm working up that I'm really excited by.
OLIVER: It's, uh, it's subtle.
CISCO: I wanted to replace the hood, but, uh, Felicity said it had sentimental value.