Learn English – “There is some evidence” or “there are some evidence”

As @xxxxxx correctly points out in their answer, a plural noun can follow no, which is termed a determiner.

For example:

1. There is no bottle in the cabinet. (s)
2. There are no bottles in the cabinet. (pl)

In the first sentence, the speaker states that the cabinet does not contain a bottle. This bottle could contain any substance, for instance it might be medicine, alcohol or even oil.

In the second sentence, the speaker says the cabinet doesn't contain any bottles.

If we replace no with a and some respectively, we'd get

1. There is a bottle in the cabinet
2. There are some bottles in the cabinet.

Effect is a countable noun

Ingesting medicine prescribed by a doctor should produce at least one positive effect in the patient, it should alleviate one or more symptoms of the illness.

• Physical rehabilitation has a positive effect on disability in multiple sclerosis patients

But medicines can also produce one or more negative effects, which are often referred to as
side effects

• Statin side effects can be uncomfortable but are rarely dangerous.

Likewise, if we changed the OP's first example, we'd obtain the following

• I noticed no effect (I was unaffected / the substance didn't produce the effect I was expecting)
• I noticed an effect (I was affected / the substance produced an effect)

and in its plural form

• I still felt no effects = I still didn't feel any effects.
• I still felt some effects = the two or more effects of the substance continued.

No!

The subject word that defines the verb here is evidence.

And the evidence validates regardless of what kind of evidence it is or from how many sources it comes.

It's a bit like saying:

Our love is based on intellectual, emotional and sometimes sexual attraction that forms a strong foundation for the future.

It's the attraction of various kinds that requires the singular verb.

However, you could use the plural if you changed your sentence to read:

The diagnosis would be based on clinical and biochemical analysis and in some cases, in-vitro evidence that validate the disease status.

Here the plural validate refers back to both analysis and evidence.