How many formula units of lithium nitrate are produced if 4.5 moles of lithium hydroxide react. BE: Fe(NO3) + 3LiOH => Fe(OH)3 + 3LiNO3?

Fe(NO₃) + 3LiOH → Fe(OH)₃ + 3LiNO₃
Mole ratio LiOH : LiNO₃ = 1 : 1
No. of moles LiNO₃ produced = 4.5 mol

1 mole of LiNO₃ contains 6.02 × 10²³ formula units.
No. of formula units = (6.02 × 10²³) × 4.5 = 2.71 × 10²⁴

Your balanced equation has one mistake in it : it should read: Fe(NO3)3 + 3LiOH => Fe(OH)3 + 3LiNO3 ... you left out the subscript on Fe(NO3)3.**
Your coefficients provide the answer to your question, they give you the ratios of reactants and products. 3 units of LiOH were used to produce 3 units of LiNO3; this is a 1:1 ratio. Therefore, we know that 4.5 moles of LiNO3 are produced when 4.5 moles of LiOH react. Since your question is asking for formula units, I suppose you have to take the answer a step further and multiply 4.5 moles LiOH x Avogadro's number (6.02 x 10^23). This would tell you the number of molecules of LiOH that are contained in 4.5 moles of LiOH.

** NO3 has a - 1 charge, just like OH, Fe has a +3; you need a subscript of 3 on Fe(NO3)3 in order to make your charges balance out. The mistake may have been a typo. :)
Good luck to you !

hthh