Notification script | from RSS to Email | Bash
I see a number of things that may help you improve your code.

Use "shebang" line

The shebang is the line at the beginning of a shell script that tells which program to use.  In this case, you probably want this:

#! /usr/bin/env bash


See this question for details.

Consider using cron instead of sleep

If this is something you want to run automatically, consider running it as a cron tab instead of using sleep within the script.

Include some comments

The program requires rsstail, mutt and sleep which is a requirement that should be documented in a comment.

Be cautious about handing variables to programs

The mutt program, like many Linux programs, has a -- option which specifies that no further options are on the command line.  This prevents the contents of $TO_EMAIL in a line like the following from being misinterpreted as a command line option.

mutt -s $TITLE -- $TO_EMAIL &lt; $BODYTEXT


Combine strings

The echo is used twice with an identical string.  An alternative approach is 

TITLE="Nuovo Articolo BDO"
BODYTEXT="New article published on the site. TITLE: $TITLE_ARTICLE - LINK: $LINK_ARTICLE"
mutt -s $TITLE -- $TO_EMAIL &lt; $BODYTEXT
echo $BODYTEXT


Avoid creating extraneous variables

Instead of creating SENDER_EMAIL, you could just specify EMAIL and then the reassignment of the latter variable before mutt is called would not be necessary.

Consider writing a portable script

By sticking closely with Posix and avoiding bashisms your code could run on many different kinds of systems, including recent versions of Ubuntu which don't use bash.