No. Your career wont be affected. The only limit you would be face is the subject knowledge. A student opting for masters in biotechnology is expected to know about chemistry maths and biology. Even in the IIT JAM paper you can opt for MSc in biotechnology even with your zoology honours.
so if you want to pursue a career in biotechnology go for it there would be no repercussions.
Most masters exam don't consider your honours subject. So basically for a career in biotechnology you don't need to be necessary a bachelor in biotechnology.
Hope I cleared your doubt.
All the best for your endeavour.