[RPG] Does a Pyromancy Apprentice with the Burn Everything Feat ignore Fire Immunity

No, with one exception.

In general, shields are not weapons. Note that in PHB1, shields are listed with armor on page 214, rather than with weapons on pages 218 to 219. The Master at Arms feat only allows you to quick draw weapons, and shields are not weapons...

...Unless it's a spiked shield, which counts as both a light shield and a weapon. Sure it doesn't make much sense to strap a shield to your arm that fast, or unstrap it, but I have trouble believing someone could realistically stow a greatbow and draw a mordenkrad in a single minor action either. Master at Arms says you can do both, though.

This actually does work together. Note the text:

You gain a +1 feat bonus to attack rolls you make with any weapon with which you have proficiency and with a wand or another item designated as a bard implement.

The first part gives you a bonus to attack rolls you make with any weapon with which you have proficiency. The second gives you bonuses with wands or other bard implements.

Note that it doesn't say "weapon attack rolls." Instead, it says "attack rolls you make with any weapon." They're not the same.

Now, the rules for Weapons as Implements and vice versa:

Using a Weapon as an Implement: If an adventurer is able to use a weapon as an implement, the weapon works like a normal implement, but the adventurer uses neither the weapon’s proficiency bonus nor its nonmagical weapon properties with his or her implement powers.... When an adventurer uses a magic version of the weapon as an implement, he or she can use the magic weapon’s enhancement bonus, critical hit effects, properties, and powers. However, some magic weapons have properties and powers that work only with weapon powers (Rules Compendium 275).

You're making an attack with a weapon with which you have proficiency. You happen to be using it as an implement, but that doesn't matter- the text says you get the bonus with attack rolls made with any weapon, and you're making an attack roll with a weapon.

This issue comes up in a variety of ways in the game. For example, if you're using a Frost Shortbow, you could attach a Siberys Shard of Merciless Cold to it to do extra damage, which would work even when making an implement attack. If you're using a Weapon of Speed Shortbow, you make make a Ranged Basic Attack with it once per encounter as a minor action...even if the RBA is using the bow as an implement, not as a weapon.