minecraft-java-edition
When I need to move a crafting table, chest, and/or furnaces, I usually use my tools. I end up moving those items a lot as I change the layout of my place / mine from time to time. I also know mining/ digging/etc wears on that tool so:
Does picking up the those items wear on my tools?
Should I move those items by hand or use the tools?
I hate breaking my tools! I want them to last forever!
I've done some number crunching and I think I have your answer for you. These numbers assume that (a) you use the mining method in the accepted answer you link, (b) that the ores are uniformly distributed, and (c) that you're running a version of Minecraft between 1.6.0 and 1.7.3 (the chart you link hasn't been updated for 1.8 yet).
Mining for an hour with ~30 iron picks should yield you:
Similarly, using ~6 diamond picks should get:
As you can see, the iron and diamond yields are far below what goes into the picks. In other worlds, through solid stone, you operate at a loss. Caves make an enormous difference when it comes to mining efficiency.
(All this, of course, assumes my math is correct. Feel free to double-check my work in the spreadsheet.)
There are several things that could be causing this:
Proxys, if you're using a proxy it could delay some information and not others.
Magic Launcher, I have found that having Magic Launcher's console open will stop all incoming connections when it receives a Coloured message. this can be fixed by launching it without the console.
Distance, if the server you play on is physically further away, then it could severly delay information going between you and the server, especially if you have a slow internet connection, to find where the server is located go to http://geoip.flagfox.net/, type the server IP or domain name in the field that looks like this: 
Server lag, if the server's TPS(Ticks Per Second) is reduced, then it will start giving up on those with a slower connection. this can be fixed by getting a better internet provider.
Best Answer
Chests and Crafting tables only take a few seconds to break with your hands, so if you want to save on your tools that would be a good idea (although stone tools are very cheap and easy to make, and have 132 uses).
Furnaces on the other hand are not possible to "mine" without a tool (you will break it eventually, but nothing will drop), and is definitely worth one use of your pickaxe.
Using a tool on the appropriate type of block consumes one use of it, using it on something else consumes two uses. Axe for wood, pickaxe for stone, etc.