What does Chris Christie is “350 pounds of toast” mean?

The real OED has this definition as a 2002 addition:

Frequently in proleptic use, especially in you're (also I'm, we're, etc.) toast:

you (I, we, etc.) will soon be dead, in trouble, etc.

So what the writer means is simply that Chris Christie's presidential ambitions are doomed.

The usefulness of the 'toast' comparison draws on the fact that toasted bread easily morphs from being desirable (when it is properly heated) to being very undesirable (if it gets overheated and chars). At that point it becomes good for nothing. Dowd's piece subtly suggests that Christie (and by implication, his career) now stinks like burned toast when she quotes Bill Maher's remark that

he should have run for president in 2012 when he had “that new candidate smell”

Incidentally, the other type of toast (as in "He's the toast of the town") refers to the custom of communally toasting a person's health or well-being by raising a glass of a (usually alcoholic) drink, uttering some words to match the occasion, and only then putting the glass to one's lips.