We are playing an evil campaign. I am a level 5 Half Orc Paladin in service to Tharizdun, God of Destruction and Madness. I want to dual class Wizard with a strong focus on the School of Necromancy. To eventually become a Master of not only Life but also Death. My DM recommended the Oathbreaker, but I lose all Paladin powers. He agreed I could dual class, but not wear heavy armor. I don't see why not. I am new to the game, but I don't see a rule that says no heavy armor if you use magic. Plus I have spells as a Paladin and I already wear heavy armor. Can anyone explain this to me?

RAW, as long as you have proficiency with the armor type you can cast spells. But if your DM says no, then it’s no.

PS- Personally I would go Warlock over Wizard for the multiclass. Warlock and Paladin are both Charisma casters does the Multiclass works better. You can go Hexblade as a Subclass which is almost broken when paired with a paladin because of Hex Warrior, Hex, and cantrips. And Warlocks are one of the better choices for a Necromancer because of the way their spell slots work. In fact, a Hexblade/Oathbreaker makes a pretty fantastic BBE.

Correct me if I'm wrong, but I thought becoming an Oathbreaker only made you lose any subclass features you had if you started out with a different oath; any main class features you have should remain the same.

EDIT:  Also, heavy armor doesn't stop you from casting spells unless you don't have proficiency, but being an Oathbreaker should still grant you that proficiency.

You can't dual class in 5E, only multiclass. Which is kind of the same but not really, depending how you look at it. As for being able to cast spells in heavy armour, as long as you are proficient in heavy armour (which you are as a paladin) you can, RAW, cast Arcane spells. Multiclassing Paladin and Wizard is a pretty weird thing to do, though, since Int will most likely be your dump stat. How about just taking the Magic Initiate feat and choose some Necromancy spells? Or maybe go Whisper Bard and take necromancy spells for your magic secrets?

Your DM is playing with an antiquated rule set in mind. In past editions, wearing armor would impose severe penalties on arcane casters. The thought process was that wearing thick gauntlets would prevent you from doing the complex finger wiggles for somatic components. Feel free to point out to them that these restrictions no longer exist, and that a multiclass paladin/wizard can cast spells in full plate. Accept that, if he overrules you, you have to abide his ruling, but ask him if there's a way for you two to work SOMETHING out so you can get what you want. In a pinch, clerics and warlocks have subclasses and spells that can support necromancy as well.

As an identical twin, I assure you that's not how it works :P