A primary difference between these two op-amps is their input voltage range and rail-to-rail specifications.
The TLC271IP is designed and specified for single-sided operation from low(ish) voltages: From 0°C to 70°C, it can operate with input voltages between 3V and 16V, so your 3.3V supply is (barely) within the specifications. Most of the specs assume it's supplied with GND and +5V.
It's also designed such that the operable input voltage range includes the negative rail. Apply 0V to your unity gain buffer and you should get approximately 0V out.
Not so for the TL061CN! It's specified for voltage ranges like ±15V, which is nowhere near +3.3V, -0V. It's also not specified for proper operation near the rails. The following image taken from page 5 of the datasheet (the uncorrelated axes are not my fault!) shows the output voltage response. Note that the X-axis is one side of the supply voltage, that is, 14V means ±14V supply (a 28V swing) gives a 25V output range extending from roughly -12.5V to +12.5V.
![enter image description here](https://i.stack.imgur.com/ZkSci.png)
3.3V is barely specified; you need to find 3.3/2 = 1.65V which is barely if at all specified on the lower end of the scale. Because the graph is a perfect line, you should probably assume that this value is calculated and not experimentally determined. Attempting to operate this device within 1V of the rails, especially with a low supply voltage, is not likely to produce the results you want.
You need an opamp specified for low-voltage supplies and at least the lower rail (preferably both) should be within the input range.
You have wildly complicated your circuit, assuming your symbols mean what the schematic indicates.
First, an instrumentation amp would look like
![schematic](https://i.stack.imgur.com/Ervvu.png)
simulate this circuit – Schematic created using CircuitLab
You show the upper amplifier, which produces OutInv, as being connected to the same ground as R3 and R9. Under these circumstances, OutInv is forced to ground (plus the input offset of the op amp), so you could profitably get rid of the op amp. Furthermore, a gain of 100 is perfectly reasonable from a single op amp, so you could do the whole thing with 1 op amp, as George Herold suggested. A simple version would be
![schematic](https://i.stack.imgur.com/Gl6Ee.png)
simulate this circuit
This will have a nominal gain of 101. Since an LM358 has a maximum offset voltage of 7 mV, you could have an offset error of 0.7 volts.
Let's assume, though, that the two input grounds are instead a separate ground, isolated from the output ground by some common-mode voltage. In this case, you have connected the upper op amp incorrectly. R1 should be connected to the - input of both amps. However, if you do this, you'll have gain of about 220, rather than 100. The equation for gain for an instrumentation amplifier (assuming R2 equals R3, R4 equals R5, and R6 = R7) is$$G=(1+\frac{2R_2}{R_1})\frac{R_6}{R_5}$$ Alternatively, depending on the output impedance of V1, you could simply use a single op amp set up as a difference amplifier with a gain of 100. You've shown the V1 as a voltage source, so this seems perfectly reasonable.
If you're worried about a stable output, I assume you're worried about noise from the input. This is best handled with an RC filter between V1 and the input.
Best Answer
Create a mid-rail voltage point by using two 10k resistors; one connected to positive supply and one connected to ground. Join the two in the middle and connect to Vin+.
With your input disconnected you will see half the supply voltage on the output of the op-amp. Now connect your input via (say) a 10uF capacitor to Vin+. You should now see the signal repeated at the output.
To use this output signal as an ac source, use another 10uF in series with the output. If you are needing low frequencies 10uF maybe too small.
This TI document has some good circuits with and without gain for ac inputs and biasing. It's called "a single supply op-amp collection".