Specific examples of signalling pathway using logical 'OR' and 'AND'?
There are thousands of examples, here I list just a few.

1) Macrophage activation. This is a complex case with many proteins acting as AND/OR. The following paper depicts a nice scheme that helps to understand the circuit.

https://bmcsystbiol.biomedcentral.com/articles/10.1186/1752-0509-2-36

2) The Lac operon that follows the logic:

if low_glucose AND lactose:
    express(lac_genes)

if (high_glucose OR low_glucose) AND no_lacotse:
    inhibit(lac_genes)

if high_glucose AND lacotse:
    express_at_low_level(lac_genes)


https://en.wikipedia.org/wiki/Lac_operon

https://en.wikipedia.org/wiki/Synthetic_biological_circuit

3) Phosphorylation and ubiquitination pathways. For example, 


  ... proteins primed through phosphorylation by one protein
  kinase are often phosphorylated processively on the N-terminal side of
  the priming phosphate by GSK3 at a series of Ser/Thr spaced by three
  residues, with the cluster of phosphates regulating protein activity
  (e.g., glycogen synthase, β-catenin). If the two sites are
  phosphorylated by different protein kinases, then this can in
  principle provide a logical AND gate in a downstream response.


http://www.sciencedirect.com/science/article/pii/S1097276507007988

4) Neurotransmitter signaling pathways.
Figure 2 of the following paper

http://www.sciencedirect.com/science/article/pii/S0022519306003675

describes the boolean logic underlining the signaling pathway


  Tyrosine hydroxylase activates itself in this model.
  There is an “AND NOT” gate between
  tyrosine hydroxylase and COMT to activate dopamine as
  tyrosine hydroxylase and not COMT activates dopamine...
  
  ...adenylate cyclase is activated by dopamine
  receptor 1 and not by dopamine receptor 2, which has been
  represented by the “AND NOT” gate between the input nodes.
  
  ...
  
  DARPP32, is activated by protein kinase A, and
  not by calcineurin, therefore protein kinase A “AND NOT”
  calcineurin activates DARPP32.
  
  DARPP32 inhibits protein phosphatase1, which is
  represented by the “NOT” gate from DARPP32 for protein
  phosphatase1.
  ...
  The activation of glutamate receptor, needs the presence of both
  protein kinase A and the ligand, glutamate therefore, a
  “AND” gate between protein kinase A and glutamate, “AND NOT”
  protein phosphatase 1. ...


Here the link to a database of natural and artificial biological logic gates and here the article presenting the database

To conclude, I would like to highlight this work on the engineering of a biological transistor.