You will have to do some sort of buffering of your input data. Your program read line-by-line and will not handle (by design) tags spanning multiple lines.

#!/usr/bin/perl -w
use strict;

my $text = do { local $/; <DATA> }; # read entire file
print $text;

$text =~ s/<B.*?ird>[^=+]*[=+]*\B//sg; # remove unwanted stuff
print $text;

__DATA__
(Example 1)
I have the following lines:
This is some text....<Bluebird>..
MBAAAEgAAAQAB
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+

(Example 2)
but sometimes it looks like this:
This is some text....<B
luebird>..MBAAAEgAAAQAoBA
AAQKAREDSCETRTBDFS
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+
[download]

And the output is as expected:

(Example 1)
I have the following lines:
This is some text....

(Example 2)
but sometimes it looks like this:
This is some text....
[download]