How to get the Resource Management Achievement
First- play on Easy Mode.
Second- if you're going for this one, the most labor-intensive achievement of the game, you're probably something of a completionist. If so, do a run for this one after your Madhouse run- the Infinite Ammo perk will be helpful for the Boathouse fight and the Mines. It also means you'll never have to pick up the Grenade Launcher.
Third- You only get to open the inventory three times during the game to get this achievement, and there are three points in the game where you're required to open the box. This means, unlike the Walk It Off achievement, you've got no discretion about when you can access your inventory. So don't open the box unless you're
About to enter the Birthday Room
Have just left the Birthday Room (Be sure to take the Crank, the Head, and the Arm)
Have reached the swamp shack by the mines.
Those are the three chances you get (it might be technically possible to skip that last one, and dodge & weave your way unarmed through the mines, but that sounds like way more trouble than it's worth).
Don't pick up items until you need them. If you don't know exactly what you're going to use that Chem Fluid for, leave it be until you do. Likewise, place items like the dogsheads or access cards in place as soon as possible, to clear the space.
Don't pick up extra weapons- The Burner and Grenade Launcher can be superfluous, particularly on Easy Mode with Infinite Ammo. Don't weigh yourself down with them.
Leave space for story items - Many key events require you to pick up an item. Make sure you have an extra space or two for such scenarios- nothing worse than having Jack come at you with mecha-scissors and finding out you can't pick up the chainsaw until you sort through your inventory.
You can't drop Keys - You should still probably get them all - you need the Crow and Snake Keys to progress, and the Scorpion Key for the M21 Shotgun, but keep in mind you won't get those inventory slots back until you reach the Testing Area
You can drop Treasure Photos - As long as Ethan's seen them, you can drop them and still claim the depicted treasure later.