How do evangelicals interpret 1 Corinthians 10:16? Evangelical Christians do not believe in Transubstantiation. Instead, a view held by many is that the communion elements are mere symbols of Christ's body and blood (source).


  The cup of blessing that we bless, is it not a participation in the
  blood of Christ? The bread that we break, is it not a participation in
  the body of Christ? 
  
  1 Corinthians 10:16 (ESV)


How do evangelicals holding to the symbolism view interpret 1 Corinthians 10:16? What does it mean to partake in the body and blood of Christ by means of the bread and the cup?
From Barnes' Notes on the Bible:


  The more obvious signification is, that there is a sense in which it
  may be said that the cup is blessed, and that by prayer and praise it
  is set apart and rendered in some sense sacred to the purposes of
  religion. it cannot mean that the cup has undergone any physical
  change, or that the wine is anything but wine; but that it has been
  solemnly set apart to the service of religion, and by prayer and
  praise designated to be used for the purpose of commemorating the
  Saviour's love. That may be said to be blessed which is set apart to a
  sacred use (Genesis 2:3; Exodus 20:11); and in this sense the cup may
  be said to be blessed; see Luke 9:16, "And he took the five loaves and
  the two fishes, and looking up to heaven he blessed them," etc.;
  compare Genesis 14:9; Genesis 27:23, Genesis 27:33, Genesis 27:41;
  Genesis 28:1; Leviticus 9:22-23; 2 Samuel 6:18; 1 Kings 8:41.
  
  Is it not the communion of the blood of Christ? - Is it not the emblem
  by which the blood of Christ is exhibited, and the means by which our
  union through that blood is exhibited? Is it not the means by which we
  express our attachment to him as Christians; showing our union to him
  and to each other; and showing that we partake in common of the
  benefits of his blood? The main idea is, that by partaking of this cup
  they showed that they were united to him and to each other; and that
  they should regard themselves as set apart to him. We have communion
  with one κοινωνία koinōnia,) that which is in "common," that which
  pertains to all, that which evinces fellowship) when we partake
  together; when all have an equal right, and all share alike; when the
  same benefits or the same obligations are extended to all. And the
  sense here is, that Christians "partake alike" in the benefits of the
  blood of Christ; they share the same blessings; and they express this
  together, and in common, when they partake of the communion.
  
  The bread ... - In the communion. It shows, since we all partake of
  it. that we share alike in the benefits which are imparted by means of
  the body of the Redeemer. In like manner it is implied that if
  Christians should partake with idolaters in the feasts offered in
  honor of idols, that they would be regarded as partaking with them in
  the services of idols, or as united to them, and therefore such
  participation was improper.


(emphasis added by me)

The idea is that it is participating in the body and blood of Christ, not partaking of the body and blood of Christ in the sense that it's literally his body. The point of the passage is to say that when we eat the elements, we participate in their symbolism, their celebration of Christ, and their uniting the believers together. Therefore, similarly, if we participate in idol feasts, we're participating in the symbolism, celebration of the idol, and uniting that comes with it. It's a warning against participating in these things, not necessarily referencing its literal transformation into Christ's body.

EDIT: This is highlighted too by Paul in verses 25 and 26 where he says they can eat anything sold in the market (some of which had been sacrificed to idols) because “The earth is the Lord’s, and everything in it." Even though it had been sacrificed to idols, by buying it in the market they weren't participating in the idol worship it was originally consecrated for.