The unique passive on Chalice of Harmony is commonly misunderstood. It will increase mana regeneration from all sources based on how much mana the champion is missing. If your champion has 0 mana, they are missing 100% of their mana and therefore their mana regeneration is increased by 100% (i.e. the regeneration is doubled). The actual amount of additional regeneration is based on the amount of base mana regeneration. This is demonstrated with the following situations:
A champion has 8 Mp5 and is currently at 25% mana (missing 75%). Chalice will increase regen by 75% to yield 14 Mp5 (passive provides 6 additional Mp5).
A champion has 8 Mp5 and is currently at 50% mana (missing 50%). Chalice will increase regen by 50% to yield 12 Mp5 (passive provides 4 additional MP5).
A champion has 20 Mp5 and is currently at 75% mana (missing 25%). Chalice will increase regen by 25% to yield 25 Mp5 (passive provides 5 additional Mp5).
A champion has 10 Mp5 and is currently at 75% mana (missing 25%). Chalice will increase regen by 25% to yield 12.5 Mp5 (passive provides 2.5 additional Mp5).
With greater base mana regeneration, the unique passive on Chalice is more effective. Ironically, with more mana regeneration, the Chalice becomes less necessary, unless one needs it badly. However, the more mana you have, the more time this item has to be useful, as it will grant you increased mana regeneration for a longer time.
Chalice of Harmony most benefits champions who have a large need for mana, such as Karthus, Orianna, Anivia, Kassadin, and Swain. It can be used by any mana champion as a source of emergency sustain, however.
Champions who have no real use for ability power or cooldown reduction are not necessarily encouraged to buy this item over Tear of the Goddess, as Tear's upgrade Manamune offers far more relevant stats to mana-hungry AD champions than Chalice's upgrade, Athene's Unholy Grail.
The unique passive does, however, increase mana gained from a Mana Potion. That is because mana potions do not literally restore mana over the duration, but increase the user's mana regeneration for that time. That boost to mana regeneration is then increased by the Chalice's passive.