This video shows that it is possible to beat the game using only 5 echoes and warps.

• Table
• Bolder
• Bed
• Sea Urchin
• Sign

@Kraiglyndor in the comments theorize the minimum echoes needed without warps. The comment posted by the user is as follows:

If you want to beat the game without wrong warping you need:

• The table to leave the first room
• Something to weigh down the boxes (other than the boulder the shrub works, people are saying the table could work somehow)
• Some way to damage enemies
• A source of fire
• The mole to dig into the secret passage
• Two statues in the Gerudo dungeon
• Bombs
• Underwater damage
• Some way to glide in the Elden dungeon (optional)
• Ice to damage the Lanayru dungeon bosses
• Electricity to solve Faron dungeon puzzles and damage the boss
• The orb for the Eternal Forest
• The wind blower or Tornado enemy may be required if there’s sand puzzles in Gerudo
• Anti-air damage

The minimum theoretical list (assuming the tables work) with 9 entries would look something like this:

• Table
• Ignizol
• Mole
• Elephant Statue
• Hawk Statue
• Bombfish
• Ice Keese
• Electric Wizrobe
• Floating Orb