flasdfljasdfjas

But I am running into an issue with Logo not being able to handle the input current.

I'm sorry, but I don't understand this.
What do you mean with input current?

Can you more explain?

Which LOGO! have you exactly? Please write the complete order number of the LOGO!.

flasdfljasdfjas

From what I have read, the Logo is unable to use a 2 wire analog input, and must instead use a 3 wire sensor. 

Now I'm confused ... 

Your topic title says:
"Normally Open Button Input into Logo"

Now you write it is a 2 wire analog input.

The terminals I1, I2, I7 and I8 can you use as a digital input (24 VDC) or as a analog input (0-10V).

Use you e.g. I7 as a analog input, then can't you use it at the same time as a digital input.  

Please upload your program as a ZIP file.

It's better to work with the original diagram as with a screenshot.

flasdfljasdfjas

Am I able to use the limit switches and the start button directly into LOGO, or will I have to do some conditioning?

Yes, you can use the button and the limit switches directly without any conditioning.

Your program doesn't works good.

If the limit switches doesn't open, the program works in a cycle without noticing that the limit switches always remain closed.