Do I have to run new wire when replacing my electric oven?
10/3 copper NM is rated for 30A under the NEC, so the old oven was more than adequately wired, and it will support the new oven without difficulty as well.

If you are of the unlucky type to have aluminum wire in your walls -- 10/3 aluminum NM will still handle 25A, so you will be fine.

However, the load must be considered as the nameplate load (no demand factors) per NEC Table 220.55 Note 4:


  
  Branch-Circuit Load. It shall be permissible to calculate the branch-
  circuit load for one range in accordance with Table 220.55. The branch-circuit 
  load for one wall-mounted oven or one counter-mounted cooking unit shall be 
  the nameplate rating of the appliance. The branch-circuit load for a counter-
  mounted cooking unit and not more than two wall-mounted ovens, all supplied 
  from a single branch circuit and located in the same room, shall be calculated 
  by adding the nameplate rating of the individual appliances and treating this 
  total as equivalent to one range.
  


The adequacy of a 20A breaker, though, depends on whether your oven is treated as a continuous or noncontinuous load as per NEC 210.20(A):


  (A) Continuous and Noncontinuous Loads. Where a branch circuit supplies 
  continuous loads or any combination of continuous and noncontinuous loads, the 
  rating of the overcurrent device shall not be less than the noncontinuous load 
  plus 125 percent of the continuous load.


and the definition of "continuous load" in Art. 100:


  Continuous Load. A load where the maximum current is expected to continue 
  for 3 hours or more.


If your AHJ agrees with Tester and I that a residential oven is a noncontinuous load, then a 20A breaker is fine; otherwise, you'll have to stick with a 30A unit.