There are different schools of thoughts on this topic, but as @dm63 pointed out, interest rate approaching zero is not a reason for QE to lose effectiveness; it is in fact a reason for conducting QE in the first place.
My preferred narrative goes roughly as follows: In a depressionary environment, investors sell assets and move money into cash. This cuts off capital flows into risky assets and the capital market stops functioning normally. To reverse this vicious cycle, the first step is to lower short-term cash rates to make funding super cheap for those that need it and to make cash unattractive to investors holding cash. After cash return has reached zero, the next step is to conduct QE, starting with bond purchases. This lowers long-term interest rate, incentivizing investors to move money away from cash and back into risky assets (stocks, high yield, etc.). As money flows back into assets, the prices of these assets increase. Not only does this create a positive wealth effect and a virtuous cycle of risk taking, it also lowers funding costs for borrowers/issuers, allowing deleveraging to happen. But as the prices of bonds/stocks/etc. increase, their future expected returns decline. This is actually when QE loses effectiveness – when the expected returns of risky assets are very low relative to cash. When this happens, additional QE won't create as much wealth effect and can't induce investors to move into risky assets any more (if cash and stocks have the same expected return, you'd just hold cash).