If the DATE column in your FAXSENT table is of type DATE then this will cause an error since you are getting text (I am guessing) from your DateBox object and inserting it into a date field. You might need to add in a TO_DATE() call to format the text from this DateBox into a proper DATE type.
Code:
Form1->ADOQuery1->SQL->Clear();
Form1->ADOQuery1->SQL->Add("Insert Into faxSent");
Form1->ADOQuery1->SQL->Add("(Date, ToPerson, ToHotel, FaxNo, msg)");
Form1->ADOQuery1->SQL->Add(" values( to_date(:Date,'YYYYMMDD'), :To, :Hotel, :faxno, :msg)");
Form1->ADOQuery1->Parameters->ParamByName("Date")->Value = DateBox->Text;
Form1->ADOQuery1->Parameters->ParamByName("To")->Value = ToBox->Text;
Form1->ADOQuery1->Parameters->ParamByName("Hotel")->Value = HotelBox->Text;
Form1->ADOQuery1->Parameters->ParamByName("faxno")->Value = FaxBox->Text;
Form1->ADOQuery1->Parameters->ParamByName("msg")->Value = msgBox->Text;
Form1->ADOQuery1->ExecSQL();
Form1->ADOQuery1->Parameters->Clear();
Replace 'YYYYMMDD' with whatever matches the expected format of the text, i.e. if DateBox->Text is "10-MAR-2006" then you would want to use 'DD-MON-YYYY' instead. If the text entered into this DateBox object isn't standardized in some way then this becomes more complicated.
Anyway, that's my guess as to what might be happening.