Arduino UNO R3 possibility
Yes, it is possible for the arduino handle communicating over serial and I2C in real time, I've done it frequently. The IR sensor should not pose a problem either. 

The ultrasonic sensor might use a timed pulse to send its information that can take a relatively long time, and the basic ultrasound code will just wait for it to finish without doing anything else. The delay itself will not cause you problems but if a lot of data is being sent to the arduino very quickly over serial it might miss some. These delays can be worked around with interrupts/some clever coding but I doubt it will ever be a problem.