This is posted in the AA forum and I thus presume that this ticket is somehow a combination of OW flights which somehow implicates AA.

Pretty much all carriers, certainly those involved in the route you describe, will cancel the remaining segments if any segment is not flown in order. Thus, if you no show for DUB-LHR, your LHR-SFO will be cancelled. Whether that cancelled segment retains any value which you could use to buy a new ticket, will depend on the fare rules for the ticket you have purchased.

Have people ever gotten away with not having a ticket cancelled? Have carriers ever rebooked without a penalty? Of course. But, you can only depend on the fare rules of the ticket which you purchased and understand that if they provide for cancellation, that is what you ought to presume will happen.

If you provide the exact details of what you have booked, including carriers and fare bases, as well as the reason you want a waiver of the rule, you will likely get a more detailed and exact answer.