What if someone who is not God's elect wants to be saved and asks God to save him?
He does not save anyone but His elect ( Romans 8:28-30, Romans 9:22-24, Acts of the Apostles 13:48 ), the "whoseover believeth", from the heart ( Romans 10:8-10 ).
Would God then reject him because he wasn't one of the elect?
Yes, He would.
See the parable of the wedding in Matthew 22.
There will be people who think that they are saved that are not.
In fact,
many people want to be saved...
and they will do whatever they think God requires in order to try and make that happen.
" Strive to enter in at the strait gate: for many, I say unto you, will seek to enter in, and shall not be able." ( Luke 13:24 )
See also Matthew 7:13-14.
They are the religious, who think that what
they do saves them, and not the Lord Himself strictly by His grace and mercy ( Romans 11:5-6, Titus 3:5-7 ).
This will apply to them:
" Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.
22 Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works?
23 And then will I profess unto them, I never knew you: depart from me, ye that work iniquity." ( Matthew 7:21-23 ).
But one thing that the Bible does not teach, is someone who sincerely desires an eternal relationship with Him and His Son that ever has that desire
apart from God changing their heart, making them born again.
Saved from Hell?
Plenty want that.
Saved
to a relationship with the God who created everything, and to praise Him for His mercy and grace towards them forever?
No one outside of those written in the Book of Life will ever truly desire that, and no one who has not believed on Jesus Christ for the forgiveness of their sins and believes every word of God will ever see that.