Should one read/watch Triangle Heart before diving into the Magical Girl Lyrical Nanoha series? The question is simple - should one read the manga, watch the OVAs, play the games (personally, I don't have access to these) of Triangle Heart before diving into the Magical Girl Lyrical Nanoha series?
In my opinion, there's nothing from Triangle Heart that would apply to Nanoha, except for the backgrounds of some characters.

Magical Girl Lyrical Nanoha is based on the spin-off of Triangle Heart 3 called Triangle Heart 3 ~Lyrical Omochabako~ (Triangle Heart 3 ~Lyrical Toy Box~).

Several things are carried over, such as:


Kyouya and Miyuki Takamachi's back-stories as bodyguards for Fiasse Crystela.
Shinobu Tsukimura returns, and in the A's manga, it is hinted that her family are vampires through a shout out to Triangle Heart when Suzuka (who didn't appear in Triangle Heart) shows her inhuman strength in the dodgeball match with Fate.


However, there are changes in Magical Girl Lyrical Nanoha which deviates from Triangle Heart's continuity, most notably:


Nanoha's father Shirō is still alive. However, he was in the ICU for some time, which led to Nanoha's insecurity at the beginning of the series
Shinobu and Kyouya are dating, making Shinobu the canon romance. (In comparison, I believe Fiasse was considered canon in Triangle Heart).
Noel K. Ehrlichkeit, Shinobu's maid is mechanical in Triangle Heart, but in Magical Girl Lyrical Nanoha, she appears to be human.
Nanoha herself doesn't have magic in the main Triangle Heart series, but she is given magic in Lyrical Toy Box (which explains the title Magical Girl Lyrical Nanoha). (In my opinion, Lyrical Toy Box is a spin-off from Magical Girl Lyrical Nanoha rather than the general Triangle Heart series).


Other characters from Triangle Hearts are used as basis for original characters in Magical Girl Lyrical Nanoha:


In Triangle Heart, Chrono Harlaown is an antagonist. However, the prototype for his Enforcer uniform is a throw back to his "evil" black outfit with spikes from the game.
Alisa Lowell is used as a basis for Arisa Bunnings
The shape-shifting fox-girl Kuon becomes the "ferret" boy Yuuno Scrya
Raising Heart's Crystal is changed from a heart to a sphere. It is also shorter in Triangle Heart and more reminiscent of Sakura Kinomoto's Sealing Wand's Star Form


In Triangle Heart, Lindy Harlaown is not a member of the Time Space Administration Bureau, but instead a fairy from the Dream World. She, however, retains much of her design, including her fairy wings, which appear in Magical Girl Lyrical Nanoha when she flies.




The main point is that a lot of the minor characters' back-stories are tweaked for Magical Girl Lyrical Nanoha, while most of the main characters except for Nanoha are all original and do not relate back to Triangle Heart (Lindy and Chrono retain their appearances, but they are still different).

As noted above, the only real change that affects a main character is that Nanoha's father isn't dead, since his time in the ICU created the Nanoha we see at the start of the series (before meeting Yuuno and using magic).