I heard this during a conference call.

Problem:
“Today” is an adverb, so preceding it with the preposition “on” is incorrect.

Explanation:
“Today” already tells you when.

For example, the answer to “When will the bus arrive?” could be “The bus will arrive today.”

In contrast, “Sunday” is a proper noun, not an adverb, so the answer to “When will the bus arrive?” could be “The bus will arrive on Sunday.”

In other words, the days of the week, which are proper nouns, require the “on” preposition to tell you when, whereas “today” already tells you when.

Solution:
“It will start today.”