Would the meaning change if I replace "dismiss" with "deny" here?

In this context, to dismiss something is to set it aside, to ignore it, to disregard it. Whether it is true or not does not matter. Thus, we might have "He has lied so frequently that I dismiss both his truths and his lies."

To deny something in this context is to assert that it is not true. "I deny that I was driving dangerously."

I believe that Hot Licks’s comment is more to the point than the other answer.

“Deny” means to say that a statement is false, but, implicitly, to acknowledge that it is a valid statement that deserves (or needs) to be addressed:

Oxford Dictionary on Lexico.com:

Refuse to admit the truth of (a concept or proposition that is supported by the majority of scientific or historical evidence)
‘an anti-environmentalist campaign group that denies climate change’

American Heritage Dictionary:

To declare untrue; assert to be false:  "A senior officer denied that any sensitive documents had been stored there" (Scott Ritter). *

Collins English Dictionary:

When you deny something, you state that it is not true.
She denied both accusations.

Macmillan Dictionary:

• to say that you did not do something that someone has accused you of doing
  deny (that):  A spokesman denied that the company had acted irresponsibly.
• to say that something is not true or does not exist
  deny the existence of something:  Officials have repeatedly denied the existence of a secret report.

By contrast, “dismiss” (in this context) means to say that a statement is absurd, and not worthy of serious consideration:

Oxford Dictionary on Lexico.com:

• Treat as unworthy of serious consideration.
  ‘it would be easy to dismiss him as all brawn and no brain’
• (Law) Refuse further hearing to (a case)
  ‘the judge dismissed the case for lack of evidence’

American Heritage Dictionary:

To refuse to accept or recognize; reject:  dismissed the claim as highly improbable.

Collins English Dictionary:

If you dismiss something, you decide or say that it is not important enough for you to think about or consider.
Mr Wakeham dismissed the reports as speculation.

Macmillan Dictionary:

to refuse to accept that something might be true or important
The senator dismissed claims that he had accepted bribes.
We should not dismiss these ideas just because they are unfamiliar.
Their evidence was dismissed as completely worthless.
The theory cannot be dismissed out of hand.

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* No relation.