Is this correct grammar: "[...] cash can't be beat."

The form can't be beat isn't "ungrammatical". In X can't be Y[ed], Y is a past participle, but most people accept both beat and beaten as valid past participles. As this NGram shows, can't be beat is far more common than can't be beaten, and becoming more so every year.

One reason for this is that beat is "simple past" ("I beat him yesterday") as well as "present" ("He cries when I beat him"), and irregular verbs like this are always susceptible to shifts in usage.

With regular verbs such as "Cash can't be ignored", the past participle is always the same as simple past ("I ignored him yesterday"), so there's no scope for confusion caused by shifting usage.

Taking another irregular verb where the past participle differs from simple past, we can mimic OP's construction with, for example...

Matricide can't be forgiven.

I'm not aware of any dialects where forgave would be acceptable here. Even speakers who've never heard of the term "past participle" know when they need to use one.

It's a colloquial expression. The more grammatically correct version would be "... can't be beaten."

I write from the perspective of someone raised and educated in central New Jersey and in New York City.

The article you link to is an opinion article; the New York Times employs different editorial standards for different kinds of articles. That statement is based on pieces I've read in the paper describing changes to the editorial approach over the years. I couldn't find any such article to link to, however.

The style guide is apparently available for purchase, but an online version is not available to the general public (see http://en.wikipedia.org/wiki/The_New_York_Times_Manual_of_Style_and_Usage for more information).

The OED comments that the past participle beat is still occasionally used for beaten in all senses, but chiefly used in the sense of to overcome and in phrases like dead-beat. It’s found, for example, in the lyrics of ‘Save It, Pretty Mama’:

I've a brand of lovin'

Can't be beat;

My way of huggin'

Is sure a treat;

I suspect, however, that the NYT is attempting a bit of demotic.