First of all, this guide WILL contain spoilers. I will try to keep them to a minimum, but at the point of reading this you should realistically complete the game at least once. If you haven't and care about spoilers, please don't continue reading!
I did this along with the achievement for healing only 3 times and I also collected all the Antique Coins and Mr. Everywheres on EASY difficulty. You don't have to do it like this, but I figured that since Healing was restricted, I wouldn't need to bother collecting any Health Items, thus saving my Inventory Slots for Ammo and Story Related items. If you took my previous advice, you should know what items are and are not relevant for story progression, so only pick these items up.
You can obviously only open the box 3 times, so NEVER use the box unless it's a part of the game which is story related and you have no choice but use it!
The 1st few areas of the game aren't difficult at all, just avoid Herbs/Med kits and collect just the items needed to progress. I also advise leaving the Video Tapes as well. You shouldn't need them to know what to do anymore and leaving them in your inventory takes up a spot for more important items. I'm not sure if you can "Discard" them either, so better to just leave them. Also worth mentioning that if you too plan on getting the achievement for using only 3 Meds, you will have to use 1 Strong First Aid Med just after being attacked trying to open the Hatch in the Main House, from there you can use 2 more Healing Items wherever you wish. Also, if you are collecting Antique Coins, as soon as you get into the Yard visit the Trailer and use them all in the Cages, (you should have 8 at this point), I used all 8 in the Magnum Cage so that I was still 1 coin away from having to carry the Magnum around with me.
In the Old House, get the Bag from the 1st Floor Save Room ASAP, it's there so it just makes sense to do so (same with any other item that increases your inventory slots, but this was the most important 1 for me).
After retrieving both the Red and Blue Keycards for the Gate in the Yard, you will eventually come to a point with a Panel requiring a Password. Lucas will throw random Passwords at you which obviously don't work. After going through the door that opens, following the path round and retrieving the Password for the Panel, you will be told "no cheating and to put all your items away", or words to a similar effect. This is the 1st part where using the Box is essential, so go ahead and put everything you have into the box. From there go through the door and complete Lucas' "activity". You again, should know how to do this, just grab the candle, stand under the water to initiate Lucas' dialogue, go to the cooker to reignite the Candle, then go to the room with the balloons and type "LOSER" as the passcode for the door. Get the Valve Handle then go back to the water, place the Valve Handle in that room to the left, go back to the Cooker, reignite the Candle and finally put it on the Cake. There's really no need to pick up any more items from this area.
After you're finished you will come to another Box, again this 1 is mandatory and you will need to use it to get your weapons etc back. I personally took the Snake, Crow and Scorpion Keys not being able to remember if I would need them again. You won't need any of them, so leave them! I wasted 3 slots of Inventory carrying them around! What you MUST take is the CRANK, the RIGHT ARM as well as the HEAD which is required to make the Serum. On top of those, obviously take any weapons you want along with Ammo/Health Items.
After defeating Jack, save either Mia or Zoe with the Serum (doesn't matter which you save). Once you take control of Mia on the Wrecked Ship, this part is pretty simple as you will start with an empty Inventory, so you can pick up whatever you like and not have to worry. This is also the only part where you actually need to pick up a Videotape and watch it. Eveline will be sat in a chair telling you you need to watch it so you can remember. It shouldn't cause you trouble carrying it anyway. Although I forget where it is exactly, there is another Bag on the Ship which increases item slots. It requires a Corrosive to get into the room, but that's all I can remember. Grab it if you can. Sorry about the vague location details. EDIT: "The bag for the extra inventory slots on the ship is in the 1F Recreation Room but it needs a corrosive to gain access", credit to ninjaraiden2003 for the exact location!
Once you finish the Cutscene with the Bakers and take control of Ethan again, after a short while you will exit the Ship and enter the Salt Mines. From here after a while you will reach another Box. This is the 3rd and final time you will need it and is also very close to the end of the game. There's also nothing story related that needs picking up at this point, so go ahead and fill up your Inventory with Weapons, Ammo, Health and whatever else you want. Just make sure you still have the item "Eveline's Tissue Samples" as these you will need to progress. I also took all my Antique Coins out of the Box and used 1 to purchase the Magnum from the Cage in this room. If you have extra Antique Coins, use them in the other Cages to free the slots (personally I just carried them with me, but you can free up the slot if you wish by doing that). From here you're set, just make sure you don't visit another Box and make sure you have enough ammo to defeat the Final Boss. I killed it with a couple Flame Rounds and the Machinegun Bullets I picked up along the way.
Just a few more things, it might be worth occasionally creating a Backup save just in case you accidentally forget an item and have to use the Box again, voiding the Achievement. Better to start again from your last backup save as opposed to restarting the entire game again.
I died 9 times in my playthrough, 2 of which were the Bossfight with Jack just after creating the Serum. When I died here, I was given the option to "Manage Items" from the Death screen. Although I didn't use it, it could be a useful thing to use to maybe put ammo away to save for later on?
Finally, if your Inventory is full and you need to pick up a story related item, you can always go to your Inventory and either use or discard something to free a slot for it. I threw away so much ammo on my run to free up slots, so the above tip could help avoid this?
Hopefully this guide is of some use, I didn't want to make it too long, but at the same time didn't want to lack too much detail. If you feel it could be improved, or find mistakes or something that needs adding; I'd appreciate your feedback and obviously give credit where due. Thanks and good luck :-)