One extreme solution for speeding up mass combats is to completely ditch the you-go I-go turn-based system. Dump initiative rolls as well. Let the PCs and GM perform actions in real time with a few ground rules for regulating movement and attacks on a battle grid.
There are a number of ways this could be done. One option is a card-based system. Give each PC a deck of cards and ask them to write down a basic 1 round action on each card:
- Move up to PC's movement allowance,
- Make one attack on adjacent opponent,
- Shoot missile weapon at foe in LOS,
- Cast spell,
- Special actions.
PCs then select x cards from the deck (where perhaps x = character level + 2 + Int and Dex modifiers) and shuffle the remaining cards.
Once combat begins, the players and GM are not restricted by turns or initiative order - they just play a card from their hand to move, attack, cast a spell, shoot a missile weapon, or whatever and then move their figure on the grid and point to the enemy figure being attacked. Players are free to draw more cards from their deck & discard as long as they don't exceed the hand limit.
When attacking a player resolves the attack as per the normal rules, for example in D&D they roll to hit & then roll damage. For combat systems using opposed rolls or where saving throws are required, the attacker could roll for both attacker & defender simultaneously. Armor class and hit points for all monsters should be visible to players to keep game play moving along. The DM will still have to keep track of lost hit points, or else players can use hp tokens.
For large battles where the GM is faced with commanding dozens of creatures, it might be fair to decrease card limits in the player's hands, or allow the GM to move/attack with multiple creatures by playing a single card.
Game play doesn't pause except when the GM needs to make a ruling.
A similar system might be devised using dice instead of cards - players roll a bunch of colored dice, blue = move, red = attack, green = cast spell, etc. to determine which actions they can perform.