Modern digital electronics is made from many tiny electronic components called MOSFETs (a type of transistor). They can be switched on and off using a voltage, and a very small amount of current. They use the majority of the power when they are switching on and off; the rest of the time they use virtually no power. It this very low-power capability which allows us to make complex digital electronics.
Electronically, a MOSFET looks like a tiny capacitor, which triggers a very low-power switch, on and off. Once there is charge on the MOSFETs capacitor, the switch will stay on or off.
A modern large-scale processor, for example the type of device Intel makes, contains billions of MOSFETs, but requires less than 100W, and switches billions of times/second. So we can be confident that a MOSFET can be switched using much less than 1/10,000,000W. At 5V, that means 1/50,000,000A, or 0.000,000,02A, i.e. 0.02µA.
So, the input circuit, connected to an Arduino's pin has some protection, to protect the MOSFETs from static electricity (which could damage or destroy the capacitor), then a complementary pair of MOSFETs (called N-Chanel and P-Chanel). The input voltage switches one on, and the other off. It typically requires less than a micro-amp of current, but at a voltage close to ground or 5V. See pjc50's diagram.
So an Arduino input pin, connected to the 5V end of the resistor and LED will take a tiny amount of current, for a short time, to switch on.
If that pin is connected between the LED and resistor, at some voltage less than 5V, but more than 0V, then it might not switch fully. It might not give a reliable value of digital 0 or digital 1, and might switch high and low rapidly influenced by electrical noise in the environment.