If I currently have 700x23/25C tires on my bike, will 700x35C winter tires fit?
35c might be a bit too wide for your rims, maybe 32c would be a better choice. They may fit your wheels just fine, but they may or may not rub on the chain stays, seat stays or the brake caliper.

35c tires are not only wider, but also have a larger outside diameter than than their smaller counterparts.

If you have enough clearance on your frame &amp; calipers, you should still consider how big of a change in size you want to try.

Consult the chart below to help determine if they will fit.

From Tire Sizing Systems


  Width Considerations
  
  Although you can use practically any tire/rim combination that shares
  the same bead seat diameter, it is unwise to use widely disparate
  sizes.
  
  If you use a very narrow tire on a wide rim, you risk pinch flats and
  rim damage from road hazards.
  
  If you use a very wide tire on a narrow rim, you risk sidewall or rim
  failure. This combination causes very sloppy handling at low speeds.
  Unfortunately, current mountain-bike fashion pushes the edge of this.
  In the interest of weight saving, most current mountain bikes have
  excessively narrow rims. Such narrow rims work very poorly with wide
  tires, unless the tires are overinflated...but that defeats the
  purpose of wide tires, and puts undue stress on the rim sidewalls.
  
  Georg Boeger has kindly provided a chart showing recommended width
  combinations:
  
  Which tire fits safely on which rim?
  
   Note: This chart may err a bit
  on the side of caution. Many cyclists exceed the recommended widths
  with no problem