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Weight is one of the numerous factors weapons have, affecting how easy it is to use. Most importantly, it has an impact on a character's ability to dodge; larger, heavier weapons and tomes mean that a unit can avoid enemy attacks with less ease. While Weight has an impact on the games, it is not so significant that a character must constantly worry about it.
In many Fire Emblem games, a unit's speed is affected by the weight of the equipped weapon if the unit's Constitution is lower than the weight of the weapon. The weight of the weapon is subtracted with the unit's constitution and the difference is then subtracted with the unit's speed. For example, consider a unit holding an Iron Sword and a Steel Sword is about to attack an enemy unit. The Iron Sword weighs less than the Steel Sword so attacking with that, depending on the unit's constitution, could mean the difference in attacking once or twice, or being attacked once or twice. Therefore, attacking with a lighter, yet weaker weapon, could end up giving you a tactical advantage.
However, in Fire Emblem: Path of Radiance, Fire Emblem: Radiant Dawn, and Fire Emblem: Shadow Dragon, speed is affected if the unit's strength is lower than the weapon's weight. Since strength is now involved, most melee units will easily overcome this issue, but mages, who generally have low strength, will still suffer from this problem.
Weight is not a factor in Fire Emblem: New Mystery of the Emblem, nor in Fire Emblem Awakening or Fire Emblem Fates, although Fates does include a debuff system for higher-tiered weapons that can resemble weight mechanics.