Italian Don
You fight the city and the Irish at the same time. Do not confuse beating a rival with your own team winning.
The Don is a mirror of the Irish Boss in mechanics: knows their Mafioso, looks for the Sheriff and takes part in choosing the night target. The difference is the game identity and the particular strategy at the table, not extra secret abilities.
The first day
Do not build your game only around accusing the Irish: if it looks like a systematic campaign against one group, the city may suspect a third side. Keep an explainable position on all candidates.
The night search for the Sheriff
Check one player per night while you are alive. Look not for "the cleverest" but for the one whose decisions may rest on information that others do not have. Before the first night nobody could have received a Sheriff's check.
Strategy against the Irish
Use public votes and behaviour to look for rivals. Do not forget: the Irish Boss is looking for the Sheriff in exactly the same way, and their shot may count on any particular night with a queue you do not know.
When the Sheriff is found
Consider the risk of their future reveal, the chance of challenging their version and the chance of removing them at night. You cannot count on your own shot surely working while both gangs are alive.
After the Don dies
The Mafioso goes on playing and shooting without searching for the Sheriff. This makes keeping the Don valuable, but does not justify any risky rescue at the price of revealing the whole pair.
Typical mistakes
Overrating your own checks; ignoring the city for a duel with the Irish; thinking that every exposed Irish player brings an Italian win closer.
Information
You know your Mafioso. The Irish are unknown. Searching for the Sheriff at night does not automatically reveal the Irish.
Day play
A reasoned attack on the Irish may be believable, but does not guarantee safety for the Italians. Analyse who will be the city's next target.
Night play
Weigh the likely Sheriff, the rivals and the players who build trust. The choice of target is a decision under uncertainty.
Against a reveal
A false counter-reveal is risky: it creates a timeline that can be checked and can fall apart after new events.