I have been separated from my husband for 4 years. I started substitute teaching bringing in approx. 3-5 K a year. My husband makes approx. 250K a year. Will my income cause the IRS to tax my husband’s income more?

This is really an accounting question, not a legal question....but I can still give you some basic answers.  It sounds like your husband is concerned that any extra earnings will throw the two of you into a higher tax bracket which can result in the IRS hitting both of you with higher tax consequences if you file jointly next year.  Emphasis here is if you file jointly.  You can file separately next year.  If you do, then your incomes will not be combined. 
If you do file jointly, despite your separation, then you still have options.  If your working would throw you into a different tax bracket, then visit with a financial adviser to reduce your tax liabilities.  For example, if you are only making 3-5K annually, then you or your husband could deposit those funds (or some other funds) into an IRA and receive a corresponding deduction.  You could also return to school and receive certain education credits.  Basically, you need to set up an appointment to consult with a tax adviser regarding your goals.
The bottom line is that it is possible for you to work and to reduce tax consequences at the same time.  Considering that you are separated, you need to continue to work towards financial independence.  If you don't start working, then it will delay you getting back to full-time employment.  So, regardless of the pressure by your husband, this is a goal you need to pursue.  I do appreciate that he is concerned about his future...but you need to be equally concerned about your future.  If you can progress forward and minimize tax effects, then teaching will be a win/win, not a loss/loss.