Here are some general thoughts. Duality is thought of in the Linear Programming sense. If the Primal is to Maximizing Output, then the Dual is to Minimize Input. In other contexts in economics, Price and Output are considered duals.

Generally, when you maximize output, the lambdas are used as weights to combine (sumproduct) the inputs and outputs columns. These sum products are constrained to be less than, equal to, or greater than the (efficiency estimate times the sum products.) 

In general we view the lambdas as in  a Lagrangian framework, where we use the terms relaxing or tightening the constraint.

Perhaps you should try to work out an easy example, such as in Chapter 2 of Cooper classical work in say an Excel file to see how the LP model works for DEA. Sorry I cannot be more specific at this time.

1. In the basic CCR model of DEA, we want to maximize the sum of weighted outputs. Then why don’t we want to minimize the sum of weighted inputs in its dual form?

2. What does the variable θ mean which is very often used in the dual form (i.e what is the name of the variable)? Is it optimal efficiency score that we are interested to find out? If it is, then why do we want to minimize efficiency in the dual form? Does it make sense to minimize efficiency?

3. Can anybody please tell me the physical interpretations of the first two constraints that are used in the dual form (i.e. Σxij*λj= yro)?

4. What is the physical significance/interpretation of the decision variable(s) (λ1, λ2,...) used in the constraints of dual form? In other words, what actually λ1 or λ2 these parameters represent and what do their values represent for a particular decision making unit, when they are addressed in the efficiency reference set or benchmarks?

5. The concept of slack in DEA is not very clear to me (though I have gone through several papers and text books). In an input-oriented model, even if a DMU has 100% efficiency, it may be inefficient compared to some other efficient unit(s) as the DMU should use less input resources. Isn’t it very surprising that a DMU is unable to use its minimum resource(s) (or in other words, it is absorbing more resources or excess inputs and it has a slack), but still it is achieving 100% efficiency!!! How this becomes possible? Again, regarding the inefficient DMUs, they should use less resource, then why the slack for these units are zero?

6. Why scale efficiency or size of a DMU (i.e. too small or too large) sometimes may be the cause of inefficiency (that the CRS model comprehends or takes into account)?

Please look at this paper:

doi:10.1016/j.eswa.2014.08.028

The terminologies of "primal form" and "dual form" are really confusing. Please use "envelopment form" and "multiplier form". If the envelopment form is to be maximized (minimized), it is an output- (input-)oriented DEA model.

In envelopment form, variables are of physical activities, ie. inputs and/or outputs adjustment; while in multiplier form, variables are of shadow prices.

In Dual we want to minimize theta although it is effciency,because our efficiency is>= 1, so we want minimize it to get close to 1

I upload DEAwithCrossEfficient.pdf that is one of first chapter of DEA with CCR and Inverted DEA models and crossEficient published from Amazon.

I explain how to build LINGO SEts model and DEA.

@ S. Assani, If in the envelopment form model (dual), the target is to minimize the inputs level radially from (x to theta*x) in the input-oriented model, then why do we minimize θ, efficiency of the reference firm?

I'm analyzing some DMU data ...

the efficiencies by the input-oriented CCR are ok!

but the efficiencies by the output-oriented RCC, which should be the same, are not!

Why?