[Note: This answer will be heavily based the materials presented in the main ASOIAF books and the other companion works.]
Probably the only other "relevant" character to commit suicide was
jumped from the top of one of the towers of Starfall, called the Palestone Sword, on the cliff atop the sea. The body was never recovered.
Who was rumored to be
loved by Ned Stark. It is rumored that she committed suicide after after giving birth to a still-born child. The fan theory is that it was Ned's.
Other "important" (not common-folk or soldiers) I can find are:
- Maester Cressen in his service to Stannis attempted to kill Melisandre by poisioning a cup of wine. Once the cup is poisoned he offer to share it with her, knowing that once they both drink they both will be dead. However, with the help of her magic necklace Melisandre is able to survive.
- Gael Targaryen committed suicide by drowning herself in the Blackwater Rush. The reason for her suicide was that she had been seduced and abandoned by a traveling singer, who had left her pregnant.
- Aelora Targaryen was married to her twin brother, Prince Aelor. Aelora caused Aelor's death in a mishap, which left her mad with grief. Aelora took her own life after being attacked at a ball by three men.
- Helaena Targaryen at the age of twenty-one, jumped from her window in Maegor's Holdfast at sunset. She died on the spikes below, with her throat impaled.
- Jaehaera Targaryen reportedly threw herself from Maegor's Holdfast (akin to that of her mother, Queen Helaena Targaryen) and was impaled on the spikes of the dry moat below. She lived for a half hour in agony before her death
- The wife of Eustace Osgrey committed suicide. This made Ser Eustace the last of his line.
and then probably the most famous/imfamous was
- Aerion Targaryen aka Aerion Brightflame died screaming after drinking a cup of wildfire, believing it would transform him into a dragon.
The last one I can think of is debateable depending on what actually happened but,
Jaqen H'ghar, or at least one of the Faceless Men
drinks a poison when a debt is owed to the gods when
Arya kills Meryn Trant
This is debatable because the said character is still shown to be alive, but there is no doubt a dead body to be accounted for...