In Ticket to Ride, when completing a route do you replace the cards you used with new ones?
No. Drawing cards, both for destinations and trains, is a separate action. Finishing a route gets you points, but you need to replenish your hand manually.
The rule book says nothing about drawing cards in the section on claiming a route, if you were meant to draw cards, it would be here:
Claim a Route – The player may claim a route on the board by playing a set of Train Car cards that match the color and length of the route and then placing one of his colored trains on each space of this route. He then records his score by moving his Scoring Marker the appropriate number of spaces (see Route Scoring Table) along the Scoring Track on the board.
Or a little later on in the section about claiming routes:
To claim a route, a player must play a set of cards equal to the number of spaces in the route. A set of cards must be of the same type. Most routes require a specific type of set. For example a Blue route must be claimed using blue-colored Passenger Car cards. Some routes – those that are Gray colored – can be claimed using a set of cards of any one color.
When a route is claimed, the player places one of his plastic trains in each of the spaces of the route. All the cards in the set used to claim the route are then discarded.
A player may claim any open route on the board. He is never required to connect to any of his previously played routes. A player may only claim a maximum of one route, hence connect two adjacent cities, never more, on his turn.
Some cities are connected by Double-Routes. One player cannot claim both routes to the same cities.
The only way to draw cards is by taking the Draw Train Car Cards action for your turn instead of the Claim a Route action:
Draw Train Car Cards – The player may draw 2 Train Car cards. He may take any one of the face-up cards or he may draw the top card from the deck (this is a blind draw). If he draws a face up card, he immediately turns a replacement card face-up from the deck. He then draws his second card, either from the face up cards or from the top of the deck. (See Train Car Cards for special rules for Locomotive cards).