Working with a system of differential equations that cannot be solved explicitly

There are many ways which one could exploit. For example let's use calculate o1''[t] eliminating o1'[t], o2'[t], o3'[t]. We can use Eliminate:

#1/.( Eliminate[ Join[ D[Halp, t], Halp], {#2, #3}] //ToRules)&[o1''[t], o2''[t], o3''[t]]

2 o1[t] o2[t]^2 + 2 o1[t] o2[t] o3[t] - 2 o2[t]^2 o3[t] + 2 o1[t] o3[t]^2 - 2 o2[t] o3[t]^2 

Changing the order in the square bracket we can calculate o2''[t] and o3''[t].

For the third order derivative of e.g. o1[t] we can eliminate other dependent variables in many different ways, let's point out one of them:

#1 /.( 
Eliminate[ Join[ D[Halp, {t, 2}], D[Halp, t], Halp], {##2}] 
//ToRules) &[o1'''[t], o2'''[t], o3'''[t], o2''[t], o3''[t], o1''[t]]

  2 o1[t]^2 o2[t]^2 + 4 o1[t] o2[t]^3 - 4 o1[t]^2 o2[t] o3[t] + 8 o1[t] o2[t]^2 o3[t]
- 4 o2[t]^3 o3[t] + 2 o1[t]^2 o3[t]^2 + 8 o1[t] o2[t] o3[t]^2 - 10 o2[t]^2 o3[t]^2 
+ 4 o1[t] o3[t]^3 - 4 o2[t] o3[t]^3  

In the above we eliminated all dependent variables starting from the second position in the square bracket (note very useful sign ##n i.e. SlotSequence ) and calculated o1'''[t]. Of course you can try to get rid of different dependent variables. In general this is a difficult issue to decide which dependent variables could be eliminated and how they could be represented by another variables. I suggest to take a closer look at GroebnerBasis with the MonomialOrder -> EliminationOrder option.

GroebnerBasis[ polys, vars, elims, MonomialOrder -> EliminationOrder]

If given equations are relatively simple you can't observe a common problem in differential elimination - so called intermediate expresssion swell. For interesting mathematical issues related to the problem I'd suggest to study e.g. a recent monograph Involution by Werner Seiler (Springer 2009). However I couldn't mention interesting packages in Mathematica related to differential elimination (there are such packages e.g. in Maple).

I would simply define rules for derivatives of o1, o2 and o3 and avoid the use of Eliminate or GroebnerBasis:

o1' = Function[t, o1[t]*o2[t]+o1[t]*o3[t]-o2[t]*o3[t]];
o2' = Function[t, o1[t]*o2[t]+o2[t]*o3[t]-o1[t]*o3[t]];
o3' = Function[t, o1[t]*o3[t]+o2[t]*o3[t]-o1[t]*o2[t]];

Derivative[n_?Positive][o1] := simplifyFunction @ Derivative[n-1][o1']
Derivative[n_?Positive][o2] := simplifyFunction @ Derivative[n-1][o2']
Derivative[n_?Positive][o3] := simplifyFunction @ Derivative[n-1][o3']

simplifyFunction[HoldPattern@Function[t_, body_]] := Function[t, Evaluate @ Simplify @ body]

Example:

o1'''[t]
o2'''[t]
o3'''[t]

2 o1[t]^2 (o2[t] - o3[t])^2 - 2 o2[t] o3[t] (2 o2[t]^2 + 5 o2[t] o3[t] + 2 o3[t]^2) + 4 o1[t] (o2[t]^3 + 2 o2[t]^2 o3[t] + 2 o2[t] o3[t]^2 + o3[t]^3)

2 (2 o1[t]^3 (o2[t] - o3[t]) - 2 o1[t] (o2[t] - o3[t])^2 o3[t] + o2[t] o3[t]^2 (o2[t] + 2 o3[t]) + o1[t]^2 (o2[t]^2 + 4 o2[t] o3[t] - 5 o3[t]^2))

2 (-2 o1[t]^3 (o2[t] - o3[t]) - 2 o1[t] o2[t] (o2[t] - o3[t])^2 + o2[t]^2 o3[t] (2 o2[t] + o3[t]) + o1[t]^2 (-5 o2[t]^2 + 4 o2[t] o3[t] + o3[t]^2))