Not a single regex but one way to do what you want is to read your file in paragraph mode, with $/ and split the logic:

{
  local $/ = ""; # Edit: added use of local for good practice

  while (<DATA>)
  {
    s/^##.*//s unless /^\w/m;
    print;
  }
}
__DATA__
## Remove

## Keep
this

## Remove

## Also keep
that
[download]

Otherwise you could use a negative look ahead assertion (?!^\w)

Edit: seems like I really didn't read the requirement well enough ^^"

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11119658
use warnings;

$_ = <<END;
## This gets stripped

## This doesn't
Because it contains this line

## More headers
blah blah

## this also should be stripped ?

## This should not get stripped, but it does

### This should prevent it from getting stripped, but it doesn't
stuff

#### This should also not get stripped

##### But it does
isn't some text needed here?
END

s/^(#+).*\n\n(?!^\1#)//gm;

print;
[download]

Outputs:

## This doesn't
Because it contains this line

## More headers
blah blah

## This should not get stripped, but it does

### This should prevent it from getting stripped, but it doesn't
stuff

#### This should also not get stripped

##### But it does
isn't some text needed here?
[download]

I think a larger test case may be needed...

$input_text =~ s{
    ^ ([#]{2,5}) \s+ [^\n]+ \s+ (?= ^ \1 [^#])
    }
    {}xmsg;
[download]

All headers that are followed by only whitespace, with the whitespace getting followed by a header with the same or fewer pounds signs as the initial header, should get stripped.

$PM = "Perl Monk's";
$MCF = "Most Clueless Friar Abbot Bishop Pontiff Deacon Curate Priest Vicar";
$nysus = $PM . ' ' . $MCF;
Click here if you love Perl Monks

use strict;
use warnings;



local $/ = "\n##"; #record separator

while (<DATA>) {
    chomp;

    #print "\n<<<$_>>>\n"; # check input

    my ($head,$rest) = /^ (.*?) \n (.*) $/xs;

    # print record only if $rest contains alphanumerics
    print "##$_" if $rest =~ /\w/;
}




__DATA__
## This gets stripped



## This doesn't

Because it contains this line



## More headers

blah blah


## This should not get stripped, but it does



### This should prevent it from getting stripped, but it doesn't

stuff



#### This should also not get stripped



##### But it does
[download]

C:/Perl_524/bin\perl.exe -w d:/exp/pm_headers.pl 
## This doesn't

Because it contains this line


## More headers

blah blah

### This should prevent it from getting stripped, but it doesn't

stuff



Compilation finished at Wed Jul 22 17:59:57
[download]

I came up with this, which is faintly ridiculous:

$input_text =~ s/^##\s+[^\n]+\s+(?=^##[^#])//gsm;
$input_text =~ s/^###\s+[^\n]+\s+(?=^###[^#])//gsm;
$input_text =~ s/^####\s+[^\n]+\s+(?=^####[^#])//gsm;
$input_text =~ s/^#####\s+[^\n]+\s+(?=^#####[^#])//gsm;
[download]

I just googled "conditional regular expression" and it seems that is a thing and it might help me. Not sure yet.