Can extension cancel the existing standard requirements?

The Standard requires that if a program violates a constraint which is in a constraints section, an implementation must issue at least one diagnostic. There is no requirement as to whether the document be meaningful or have any relation to the constraint violation. An implementation that unconditionally output "Warning: this implementation makes no attempt to enforce constraints its author views as stupid" would suffice. Likewise an implementation that includes a command-line option to output such a message and documents that it may not be conforming unless that option is specified.

Note that even that requirement has a loophole: if a program exceeds an implementation's translation limits, the implementation may behave in any manner whatsoever, without limitation, and without having to issue any sort of diagnostic. Although the Standard requires for each implementation there must exist at least one source program that at least nominally exercises the translation limits given in the Standard without causing the implementation to malfunction, an implementation may impose arbitrary restrictions on how the translation limits interact, e.g. allowing a program to either contain one identifier of up to 63 character, or a larger number of identifiers that are no more than three characters long. There are very few circumstances where anything an otherwise-conforming implementation might do with a particular source text would render it non-conforming.

It's not so much that an implementation "cancels" a requirement with an extension, but that extensions add features that the standard doesn't otherwise support. The main requirement is that extensions doesn't alter any strictly conforming programs.

The definition of a conforming implementation is as follows from section 4p6 of the C11 standard:

The two forms of conforming implementation are hosted and freestanding. A conforming hosted implementation shall accept any strictly conforming program. A conforming freestanding implementation shall accept any strictly conforming program in which the use of the features specified in the library clause (clause 7) is confined to the contents of the standard headers [ ... omitted for brevity ... ]. A conforming implementation may have extensions (including additional library functions), provided they do not alter the behavior of any strictly conforming program

Where a strictly conforming program is defined in section 4p5:

A strictly conforming program shall use only those features of the language and library specified in this International Standard. It shall not produce output dependent on any unspecified, undefined, or implementation-defined behavior, and shall not exceed any minimum implementation limit

And a conforming program is defined in section 4p7:

A conforming program is one that is acceptable to a conforming implementation.

So given the case of the program from your prior question:

static int arr[ ];

int main( void )
{
        return arr[ 0 ];
}

static int arr[ ] = { 0 };

This is not a strictly conforming program because it violates 6.9.2p3. However some implementations such as gcc allows this as an extension. Supporting such a feature doesn't prevent a similar strictly conforming program such as this

static int arr[1];

int main( void )
{
        return arr[ 0 ];
}

static int arr[ ] = { 0 };

From behaving any differently. Therefore an implementation supporting this feature still qualifies as a conforming implementation. This also means that the first program, while not a strictly conforming program, is a conforming program because it will run in a well defined manner on a conforming implementation.