You need to ask the lawyer to show the contract between you and her (or her office) regarding the investigating the title to the property. The lawyer should be itemising the bill and the title investigation itself should only cost $16.35 (& her bill should show this).
I made an offer to a house, before signing the offer, I faxed the offer to a lawyer and she made some suggestions of the offer before I signed it
She can charge you for perusing the contract - not all lawyers do it for free as it is their billable time, and aparently you agreed with it by signing. You can argue with her on the point of the titles search, but by getting her to look at the contract, your rendering of her service was inferred. Of course you should never make assumptions and talk upfront about fees/costs.
she is asking me to pay for perusing draft contract and investigating title to the property
If you & the lawyer don't have a written contract, and her charging you was inferred by you asking her to look at the contract, you can argue this as all you requested was a contract of offer. The lawyer didn't have explicit consent from you to do a title search so she should not charge you for it & of course not charge you for her time doing something you did not explicitly requested.
I am wondering if I should pay her or not, because I did not ask her to do the title search at all, and some lawyers do not charge for reading draft contract.
Personally I would pay for doing the drafting of the contract - you will lose on this point in front of a judge as you signed it after she made suggestions - so you agreed to her work, and I would request a please explain on the title search and get her to itemise the bill including down to the minute, and use of anything for the completion of said service - just cause I can. & then based on her explanation(s) will act accordingly (don't know if its worth fighting, but hopefully at least she realise she jumped the gun and decrease the bill).
What's the possible result of ignoring of this bill letter?
Well I believe its better to acknowledge such bills and letters when it comes to financial matters, the sooner the better. I would write a letter back to the lawyer acknowledging you have received their letter on such date and acknowledge her bill. Point out that you did ask her to draft up a letter of offer to the vendor on such date but did not explicitly request a title search. Point out that if was just an offer and because a binding contract between you and the vendor was not yet in place, a title search on the property would be jumping the gun. (I had my conveyancing, finance & inspections lined up and ready to go, but I would not go ahead til I saw the vendor's signature on the contract).
THIS IS IMPORTANT - KEEP COPIES OF ALL LETTERS RECEIVED & SENT - WHEN YOU SEND A LETTER BY MAIL, ALSO SEND A FAX & KEEP THE FAX CONFIRMATION, DATED WOULD BE GREAT. IF YOU NEED TO SPEAK ON THE PHONE, KEEP A RECORD OF THE CONVERSATION AND ACKNOWLEDGE THE CONVERSATION VIA MAIL/EMAIL/FAX - if you had to go to court, at least you have records of the story.
Now what happens if you ignore the bill or choose not to pay up? Well it depends on how big the bill is. I'm gonna assume its a few hundred dollars to maximum of a few grand. I would consider it a minor debt, and the smart thing for the lawyer to do (that is if you don't pay up) is to take a minor debt claim against you, in which you have 28 days to defend the claim. You won't need a lawyer for this - you can represent yourself. Worst case it goes before the magistrates and you are enforced to pay the money - of course the court will work with you to set up a payment plan to pay the debt back, you can get more information via
http://www.legalaid.qld.gov.au/Publications/Factsheets+and+guides/Self-help+kits/Minor+debt/
You can also ask legal aid for free advice.