There are many items that have various effects tied to your constitution modifier. I have chosen only to list those items for which a negative constitution modifier creates a serious malfunction.
Ring of Temporal Salvation
If you die while wearing this gray crystal ring, you vanish and reappear in an unoccupied space within 5 feet of the space you left (or the nearest unoccupied space). You have a number of hit points equal to 3d6 + your Constitution modifier. If your hit point maximum is lower than the number of hit points you regain, your hit point maximum rises to a similar amount. If you have any levels of exhaustion, reduce your level of exhaustion by 1. Once the ring is used, it turns to dust and is destroyed.
A negative constitution modifier can nullify the effects of this magic item entirely. With a modifier of -3 or lower, rolling three 1s on the 3d6 means you reappear unconscious.
A non-evil creature can carry a number of soul coins equal to or less than its Constitution modifier without penalty. A non-evil creature carrying a number of soul coins greater than its Constitution modifier has disadvantage on its attack rolls, ability checks, and saving throws.
With a constitution modifier less than or equal to 0, a single soul coin imposes disadvantage on all attack rolls, ability checks, and saving throws.