- Even a small noise on IN+ causes some current to pass through Rsense and make the LED turn on, although it is very weak, it consumes power and not ideal. How can I make this work only above certain voltage on IN+, for example 1V.
Simply take your feedback from the output of the op-amp (the base of the BJT). Now if the input signal is less than about 0.6 V, no IR output.
If you still want to have the same maximum output you can adjust the two resistors (not labelled in your schematic) to increase the "on" voltage input to the op-amp by 0.7 V or so.
As Nick Alexeev points out the above is a bit of a "quick-and-dirty" solution. It gives about 100-200 mA of uncertainty of the output current. It might also restrict your options with solving the issue of the op-amp not having enough "oomph" to provide the base current in the BJT that you need to get the output current you want (as detailed in Nick's answer).
Here's a solution that solves both problems (this replaces the op-amp, BJT, and sense resistor in your circuit):
With this circuit you don't need to worry about the drive capability of the op-amp. You also reduce the output current uncertainty. The original circuit had possible errors due to the resistor variation (0.5 - 5%, depending what resistor you buy) and the BJT beta (1 - 2%).
With this you are left with just the resistor variation and an error due to the variation in the diode forward voltage.
If the 47 kOhm resistor makes this circuit respond too slowly for your needs, you may need to reduce the values of your voltage divider resistors to enable you to reduce the value of the new resistor.
To reduce component count, you could build the diode in to your resistor divider. R1 here replaces the lower resistor in the divider. The value of R1 would be reduced to 10 kOhms. An additional resistor is added in series with the diode to be the upper resistor in the divider. The input then comes straight from the 555.
The value of the upper resistor should be chosen to give the output current you want when the input is high.