Here's a list of bugs I found so far while playing for about 2 hours. I'll add new bugs I find to this list later.
Zamorak teleport brings you outside the boss room, and clicking the door doesn't give access to the room either.
The spades at Barrows don't work.
The first Agility shortcut in the 'Dragon Slayer Dungeon' doesn't work. The animation will occur, but your character then teleports back to the Mithril Dragons.
Bat bones can't be buried.
Kree'arra can be killed with Melee. (credits: Casualbossin)
Super Antipoison potions do not work. (credits: Casualbossin)
Can't make Overloads. (credits: Casualbossin)
Added:
Black Demons are counted towards the Boss Kc.
When trying to select autocast for any spells using earth or water runes with the Mud Battlestaff, you get a message saying you don't have enough water runes.
So far, Melee seems extremely inaccurate at Barrows, even against Karil and Ahrim, with 99 Attack and Strength. (tested with both Dragon Scimitar and Abyssal Whip. Credits: Casualbossin)
Gargoyles only give 20.000 Slayer experience, while monsters like Blue Dragons (32.000) and Aberrant Spectres (26.500) for example, give more exp.
Added 1/08:
Magic Staffs (Ancients, Ahrim's) only give +15 Magic bonus, but much more crush and strength bonusses, I assume this needs to be fixed.
Construction is very buggy. When the home portal is entered, ::home brings you to Ooglog, and if you try to build a door, you remove your garden and can't train anymore.
Bandos Helm has no stats.