For example, if you were to put 5V across a \$150\Omega\$ resistor and measure its resistance with a multimeter, the reading would always be a few percent off the true value. What reasons might there be for this inaccuracy?
The resistor may have a 1% or 2% tolerance (5% resistors used to exist, but are pretty rare these days, at least for professional suppliers).
Then check the specification of your meter. I would be surprised to find it had a better accuracy than 2% unless it was a really expensive one.
These tolerances can add up worst case, to give you 3% to 4% error, with meter and resistor fully inside specification.
The connection between probes has a finite resistance that needs to be zeroed out. Touch your probes together to see if you get zero volts. If you don't, some meters have a way to zero this out, or you can do it in your head.
At 150 ohms, 3 or 4 ohms in series in the probes/meter is a significant percentage.
Of course, you need to remove the resistor from the circuit for accurate measurement. If you are measuring HIGH resistances, you also can't hold the ends of the resistor in your hand, as your body impedance in parallel can impact the measurement.
The multimeter itself is actually correct because all resistors have a tolerance (so the resistance of the resistor could be +/- a certain percentage).
The multimeter measures much more accurately and gives you the real value of the resistor.
You can get more accurate resistors if you need however for simple, 'normal' uses, a normal resistor with a 1%-5% tolerance should be fine.
Commercially available resistors will typically be 5% or 1% tolerance, which means that within their operating temperature range, they will deviate no more than that much from their nominal value.
When you put some voltage across a resistor, it will conduct current and start dissipating some heat. This will cause a change in temperature, which will in turn change the actual resistance, but should still keep it in tolerance range.
On a side note, Your multimeter's manual suggests to remove components from circuit board while measuring their attributes. If a component is within a powered circuit and you attempt to measure resistance/capacitance/hFe etc, it may give widely incorrect readings, and may also kill your multimeter.