Death in the Family isn't what I'd call a good story. There are a lot of moments which don't make sense especially when you think past the "that's weird" bit. This list talks about things that have no logic, are dumb and a few big plot holes.
Honorable Mentions: Joker pointlessly trying to hide the evidence. Wouldn't he want Batman to know? Joker even taunted Batman in the Batman Under the Red Hood movie when asked about Red Hood saying something to the affect of "I'd want you to know it was me."
Bruce referring to Jason by name. Much later in another story he actually says "Jason Todd" while punching Joker. What the hell?
The "Diplomatic" Part: I don't know why we had to have yet another diplomatic immunity plot in a Jason Todd story. Especially one that's rendered meaningless and a waste of time. The whole thing can easily be removed from the story. It's completely nonsensical to make Joker--JOKER--an ambassador of any country.
Bruce Wayne doesn't understand trauma: There was a plot before Jason died that dealt with a woman being raped by a man named Felipe that escaped justice because of his fathers' diplomatic immunity. Long story short it greatly affected Jason. Some speculated that he was assaulted himself (*1) if not he likely would have known someone that had been. Anyone remember what Batman and Robin were doing at the beginning of DITF? Taking down a kiddie porn ring. It NEVER occurs to Batman that might possibly be part of the reason Jason is distressed?!
Forgetting Justice League International: They could have done so much and make more sense than Superman showing up. Not just to deal with the BS at the end of the story but to help out in general. You'd think this situation would seem a little familiar too. Booster lost his sister at the end of his first series around this time (*2) when she ran off to discover America/the Goldstar costume. Which leads to my next complaint.
Bruce being Choosing the Joker over Jason: Make no mistake Bruce is a terrible father in this but this one defies logic. Joker is loose and Jason just ran away. What should he do? The book makes a big deal about how hard the choice is but Joker is who he chooses. Hey, here's an idea. Get over your ego and let the JLI or Superman handle Joker. A parent goes after their kid especially a kid he earlier feared was SUICIDAL ! You may ask why not send other heroes after Jason but that would a) Prove Bruce doesn't see him as important and b) Superman would call him on that BS. But by just going lone wolf to go after Joker instead Bruce is being a horrible parent.
The "Tests" of Jasons' Potential Moms: DNA testing might have been out of the question when this was written but there were other ways. Batman trusts a spy to be honest with him, while the truth serum with Shiva was better we latter learn it didn't work (*3) and Shelia Haywood was vague enough that many readers still don't believe she was the mom. Bruce believes it because she had Jasons' eyes. When she dies she comments that Jason must have loved his mother (*4) and given her criminal career it seems likely that she'd lie. Why? Well she did recognize Bruce Wayne by sight so she knows he's rich. While she knew who Jason was that doesn't confirm she's the mom either.
Bruce not understanding Jason needs him: Like I said Bruce doesn't win any awards for parenting in this one. When he finds Jason by chance after he ran away Jason tells him about his mom search. Jason is happy because he thinks their work together and is under the impression Bruce is there for him. But no, Bruce tells him that he's only there for a case. Ouch. Jason deflates and Bruce acts like that the kid being bratty. They only work together because everything in the plot just happens to link together. After Jason meets Shelia he leaves the two alone forgetting this is the woman that chose to leave her sons' life before. Bruce even implies he's leaving Jasons' future in his hands reinforcing the idea that he doesn't care.
Jason being "reckless"/"Life's a Game": The infamous "Life's a Game" line never ceases to piss me off. Jason is being mocking because he's hurting when he says it. That's when their taking down the porn ring (which has Batman tossing guys his way so he can work his anger out. Brilliant Bruce.) They very next page has Bruce telling Alfred that he thinks Jason is acting like someone looking to die. The two big moments when Jason is "reckless" has him rushing a gunman which gives the spy time to save herself and him hitting Shiva from behind. Which gives Batman the chance to get the upper hand. Both work in their favor yet Jason is presented like he's the most reckless person alive. Throughout this Jason is showing concern for others showing he does take life seriously and he's acting like a hero. He doesn't "challenge Joker" Jason tried to get his mom to safety when the coast was clear and only dies because he trusted her.
Superman is useless: I'm not even talking about the fact he sides with the UN (and yes I see the parallels to DKR,) I mean the fact the guy with different types of visions that is the most powerful being in this story can't find Joker. Really? So Superman showed up just to hinder Batman and offer a short "sorry your kid died" speech ?
Willis Todds' Black Book: The address book that served as a clue how to find Jasons' mother. Sure this could be people he happened to know including co-workers but there was an implication that Willis was a player. The bit with Shelia glossing over the fact Willis left her for Catherine being part of it. Anyway it's NEVER explained how a low ranking henchman like Willis knew LADY SHIVA ! I've heard someone once defend this stating she wasn't Shiva back then. But you mean to tell me Willis happened to not only know (dated?) one of the greatest (if not greatest) fighters in the world but also a spy? REALLY?!
Jasons' Birth Certificate: This is the biggest plot hole of them all. First of all, you mean to tell me they couldn't get a printed copy of it? I'll shrug off it either not occurring to Jason because of his distressed state or him being cut off from bits in the bat computer. But Bruce Wayne should have known ! Not only because he researched Jason Todds' life (*5) but he freaking adopted him ! I'm pretty sure you'd need a copy of a birth certificate for that ! Yet Bruce acts surprised when Jason tells him, doesn't know who the mom is in his narrative and he'd be pointlessly risking Jasons' life if he did know. Does Alfred handle all the paperwork and it never occurred to him to ask who the mother listed was?! Bruce sucks as a detective in this.
*1 I think this was what Tony Daniels was alluding to in BFTC. Originally they wanted to give him AIDS but never said how he would have contracted it. While it is possible I don't like the idea of this or the forced prostitution concept being canon. The guy already has enough trauma and it's not dealt with well. People already blame him for his death inside and out of the comic. Not to mention Dick Graysons' previously had a least 2 assaults that were never addressed properly.
*2 Booster Gold #22 (Michelle's death) was November 1987 with the funeral the next month (which Batman attended.) Death in the Family #1 was December 1988. Yes, the JLI arrived in BG #22 but if they had been there sooner they could have saved her.
*3 Later on it's revealed that Shiva was Cassandra Cains' mother despite saying she had no children under truth serum. Jason and Cass were supposed to be close in age though.
*4 From a more poetic point of view Shelia could mean Catherine.
*5 By looking up Catherine after meeting Jason and later on knowing Willis' connection to Two-Face.