##JavaScript (ES6), 77 58 57 56 bytes
f=s=>f=s=>s==(S=ss=s.replace(/\(\+\)|\[:]|{\|}|<->/,''))==s?!s:f(Ss)
###Test cases
f=s=>f=s=>s==(S=ss=s.replace(/\(\+\)|\[:]|{\|}|<->/,''))==s?!s:f(Ss)
console.log("Testing truthy values");
console.log(f("(+)(+)(+)"));
console.log(f("[[[:]:]:(+(+))]{<->|<(+)->[:]}(+)"));
console.log(f("<<<<<<<<<<<<->->->->->->->->->->->->"));
console.log(f("{|(+[:<-{|(+[:<->])}>])}"));
console.log(f("[[[:]:[:]]:[[:]:[:]]]"));
console.log(f("{[:](+)|<->{|}}[(+)<->:{|}(+)]"));
console.log("Testing falsy values");
console.log(f(":["));
console.log(f("<|>"));
console.log(f("(+(+)"));
console.log(f("[:][:](+[[:]):]"));
console.log(f("{|{|{|(+{|{|{|}}}}}+)}[:]{|}"));
console.log(f("{{||}}"));
console.log(f("<<->-<->-<->>"));
console.log(f("[(+):((+)+)+(+(+))]"));
console.log(f("<<<<<->->->->->->"));