Does human effort oppose God's grace? A pastor in my church once told me that "human effort does not oppose God's grace", I.E. to say, as long as you do your best (which is impossible to attain, but you try any way), God will indeed do the rest.

Are there any biblical references that support/reject this view?
Human effort consistent with the will of God, does not oppose God's Grace. However, the words of Jesus to Peter (cf. Matthew 16:21-23), 


  "Get behind me, Satan! You are a stumbling block to me; for you are setting your mind not on divine things, but on human things" (NRSV) 


are a proclamation from God himself that human effort can, indeed, oppose God's grace.

The assertion that one's best is impossible to attain is a logical paradox, and a fallacy. Everyone can do his or her best, so it is obviously possible to attain. The issue is the extent to which the "best" of any particular individual comes close to God's objective standards. But God exhorts in Genesis 4:7 


  "If you do well, will it not be accepted? And if you do not do well, sin is lurking at the door; its desire is for you, but you must master it." (NRSV)