Yes, the chromatic number of a Hasse diagram can be arbitrarily large. Bollobás shows in the following article that for any $k$ there exists a finite lattice whose Hasse diagram is not $k$-colorable.

MR505730 05C15 (06A20) 
Bollobás, Béla Colouring lattices. Algebra Universalis 7 (1977), no. 3, 313–314.

Yes, the chromatic number of a Hasse diagram can be arbitrarily large. Bollobás shows in the following article that for any $k$ there exists a finite lattice whose Hasse diagram is not $k$-colorable.

MR505730 05C15 (06A20) 
Bollobás, Béla Colouring lattices. Algebra Universalis 7 (1977), no. 3, 313–314.

Here are a few other papers that solve the problem, but I have not been able to find any freely available paper.

MR748915 05A17 (06B99 06D99) 
Nešetřil, Jaroslav; Rödl, Vojtěch Combinatorial partitions of finite posets and lattices—Ramsey lattices. Algebra Universalis 19 (1984), no. 1, 106–119.

MR1129613 05C15 (06A07) 
Kříž, Igor; Nešetřil, Jaroslav Chromatic number of Hasse diagrams, eyebrows and dimension. Order 8 (1991), no. 1, 41–48.